Taylor Series Research Articles

A Linearization based on Taylor Expansion to Multi-Objective Linear Fractional Program with Fuzzy Coefficients and Fuzzy Decision Variables

A Linearization based on Taylor Expansion to Multi-Objective Linear Fractional Program with Fuzzy Coefficients and Fuzzy Decision Variables

Modularity of Taylor coefficients of non-holomorphic Jacobi forms arising from partitions

It is well-known that Taylor coefficient of holomorphic Jacobi forms is quasimodular forms, and recently it was proved by Bringmann that such a property still holds for certain non-holomorphic Jacobi forms arising in combinatorics. In this paper, we prove further modularity results of an infinite family of combinatorial sums related to non-holomorphic Jacobi forms. More precisely, for each k≥2 and ℓ≥1 we find non-holomorphic modular forms r2ℓ−1,k(τ) arising from partitions. It turns out that the non-holomorphic part of r1,k(τ) is related to the generalized Rogers-Ramanujan functions. This appearance of generalized Rogers-Ramanujan functions is supported by the fact that this r1,k(τ) is a mixed harmonic Maass form.

A Note on Confidence Intervals for a Binomial p: Andersson–Nerman vs. Wilson

ABSTRACTNumerous approximate confidence intervals in closed form have been suggested over the years for the parameter in the binomial distribution. One of the oldest and still most advocated is the Wilson (score) interval, which in this article will be compared with the less well‐known Andersson–Nerman (henceforth AN) interval. These intervals are quite closely related, but it will be shown analytically and illustrated by examples that the coverage probability of the AN interval always equals or exceeds that of the Wilson interval, while also having uniformly larger expected length. Asymptotic expressions for the coverage probability and expected length of the AN interval are provided, using Edgeworth and Taylor expansions, respectively. The well‐behaved Wilson and AN pivots are furthermore contrasted with the problematic Wald pivot. The latter gives rise to the Wald interval, which is probably one of the best known and most used procedures altogether in the history of statistical inference. Unfortunately, this interval performs poorly, but even to this day it is to be found in many current textbooks. Therefore, it is still of relevance to search for attractive alternatives based on sound statistical principles.

A New Class of Generalized Starlike Bi-Univalent Functions Subordinated to Legendre Polynomials

A new class of generalized starlike bi-univalent functions is introduced in this paper, which is defined using Legendre Polynomials within the open unit disk D. This paper sheds a light on the properties and behaviors of these starlike bi-univalent functions, providing estimations for the modulus of the initial Taylor series coefficients of the functions falling under this particular class and one of its various subclasses. Additionally, this paper also investigates the classical Fekete-Szegö functional problem for functions f that are part of the aforementioned class. Moreover, we obtain the classical Fekete-Szegö inequalities of functions belonging to this class and to one of its various subclasses.

Nonlinear dynamics analysis of A quad-stable electromagnetic energy harvester

This paper constructs a quad-stable system using magnets and inclined springs, and derives the magnetic force between the magnets based on the magnetic dipole hypothesis. Using the Lagrange principle, the dynamic equations of this quad-stable electromagnetic energy harvester (QEEH) are obtained. A detailed parameter analysis of the system's potential energy and restoring force is conducted, and the bifurcation of equilibrium points and dynamic bifurcation behavior under different parameters are explored. To more accurately obtain the resonant characteristics of the system, a higher-order Taylor expansion is used to expand the restoring force, noting the presence of quadratic and quartic nonlinear terms. By employing the multiscale method, an approximate solution at the equilibrium point of the QEEH system is obtained, and the amplitude-frequency characteristics of the system under different parameter conditions are studied. Experimental validation demonstrates the voltage output performance of the harvester under various parameter settings.

3D MR elastography at 0.55 T: Concomitant field effects and feasibility.

To demonstrate the feasibility of hepatic 3D MR elastography (MRE) at 0.55 T in healthy volunteers using Hadamard encoding and to study the effects of concomitant fields in the domain of MRE in general. Concomitant field effects in MRE are assessed using a Taylor series expansion and an encoding scheme is proposed to study the corresponding effects on 3D MRE at 0.55 T in numerical simulations and in phantom experiments. In addition, five healthy volunteers were enrolled and scanned at 60 Hz mechanical excitation with a Hadamard-encoded 3D MRE sequence at 0.55 T and were also scanned with a reference 3D MRE sequence at 3 T for comparison. The retrieved biomechanical parameters were the magnitude of the complex shear modulus (|G*|), the shear wave speed (Cs), and the loss modulus (G″). Comparison of apparent SNR between 3 T and 0.55 T was performed. Theoretical analysis, numerical simulations and phantom experiments demonstrated that the effects of concomitant fields in 3D MRE at 0.55 T are negligible. In the healthy volunteer experiments, the mean values of |G*|, Cs, and G″ in the liver were 2.1 ± 0.3 kPa, 1.5 ± 0.1 m/s, and 0.8 ± 0.1 kPa at 0.55 T, respectively, and 2.0 ± 0.2 kPa, 1.5 ± 0.1 m/s, and 0.9 ± 0.1 kPa at 3 T, respectively. Bland-Altman analysis demonstrated good agreement between the biomechanical parameters retrieved at 0.55 T and 3 T. A 2.1-fold relative apparent SNR decrease was observed in 3D MRE at 0.55 T in comparison with 3 T. Hepatic 3D MRE is feasible at 0.55 T, showing promising initial results in healthy volunteers.

Asymptotic Numerical Method for dynamic buckling of shell structures with follower pressure

Asymptotic Numerical Method (ANM) is applied to non-linear dynamics of thin-shells subjected to conservative and non-conservative loads such as follower pressure. ANM is decomposed into several stages: the finite element discretization of the non-linear equations of motion of the shell dynamics, a homotopy transformation of the semi-discrete non-linear equations, a perturbation technique to expand the quantities into Taylor series according to the homotopy parameter and the time integration scheme to solve the series of linear problems resulting from the perturbation technique. ANM is applied here with the 7-parameter shell elements thanks to the Enhanced Assumed Strain (EAS) concept and implicit Newmark integration. In the case of non-conservative force, follower pressure also requires to be decomposed in either Taylor series or rational Padé approximants. The academic case of the cylindrical roof with dynamic snap-through phenomenon is investigated for the purpose of comparing ANM strategies and the classical Newton–Raphson (NR) method. Two engineering cases including an I-shaped thin-walled beam and a closed thin-shell cylinder under dynamic external follower pressure are also investigated. ANM turns out to be accurate, robust and efficient in terms of computation time, providing an alternative method to the well-established Newton–Raphson method.

Optimization of solar radiation on tilted surface in isotropic and anisotropic atmospheric conditions

Solar energy is the main source of energy among all renewable energy sources and it can be harvested from various types of solar collectors and PV systems. The tilt angle of solar collectors and PV systems plays an important role in harvesting maximum energy from it. This research is concerned with the assessment of solar radiation and optimization of tilt angle at Delhi (28.61° N, 77.21° E), India. Optimization of the tilt angle has been done by two methods. The Liu & Jordan model has been used to assess the solar radiation for tilted surfaces under isotropic atmospheric conditions and the HDKR model has been used for total solar radiation assessment for tilted surfaces under anisotropic atmospheric conditions. Initially, partially differentiate HT for tilt angle and put it zero, then find out the tilt angle. After applying Taylor's series expansion method on the computed tilt angle. While applying to New Delhi, a measurable difference has been found in collected solar radiation. Assuming isotropic atmospheric conditions, a 55.3 % increment in monthly mean daily solar radiation has been found; whereas for anisotropic atmospheric conditions, an increment was evaluated by 57.7 % in December. Since monthly tilt angle adjustment is not feasible hence, this research proposed seasonal tilt angle adjustment. Maximum energy loss on seasonal adjustment is found at 4.07 % in autumn on considering isotropic atmospheric conditions. While considering anisotropic conditions a 4.51 % loss of energy is observed.

Explicit maximal totally real embeddings

This article deals with an explicit canonical construction of a maximal totally real embedding for real analytic manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent bundle or of its complexified tangent bundle. The existence of maximal totally real embeddings for real analytic manifolds is known from previous celebrated works by Bruhat-Whitney [1] and Grauert [4]. Their construction is based on the use of analytic continuation of local frames and local coordinates that are far from being canonical or explicit. As a consequence, the form of the corresponding complex structure has been a mystery since the very beginning. A quite simple recursive expression for such complex structures has been provided in the first author's work “On maximal totally real embeddings” [12]. In our series of articles we focus on the case of torsion free connections. In the present article we give a fiberwise Taylor expansion of the canonical complex structure which is expressed in terms of symmetrization of curvature monomials and a rather simple and explicit expression of the coefficients of the expansion. We explain also a rather simple geometric characterization of such canonical complex structures. Our main result and argument can be useful for the study of open questions in the theory of the embeddings in consideration such as their moduli space.

Closed-form expressions for magnetic polarizabilities of circular apertures excited by nonuniform magnetic fields

Abstract The problem of electromagnetic penetration through an aperture in a perfectly electrically conducting plane is a classical problem in electromagnetic theory and applications. This work aims to investigate the behavior of an electrically small aperture when excited by nonuniform magnetic fields. Based on the Taylor expansion of the exact solution in integral form of the corresponding mixed boundary problem, we derive the equivalent magnetic dipole moment of a circular aperture. Closed-form expressions of dipole moments are derived. Also, for the special case of axisymmetric excitation where the dipole moment vanishes, we develop a quadrupole model to describe the aperture behavior. The effectiveness of the obtained expressions is verified by comparisons with accurate analytical solutions in special cases and the exact solutions in integral form.

The breather, breather-positon, rogue wave for the reverse space–time nonlocal short pulse equation in nonzero background

In this paper, by using the Darboux transformation (DT), two types of breather solutions for the reverse space–time (RST) nonlocal short pulse equation are constructed in nonzero background: bounded and unbounded breather solutions. The degenerate DT is obtained by taking the limit of eigenvalues and performing a higher-order Taylor expansion. Then the N order breather-positon solutions are generated through degenerate DT. Some properties of the breather-positon solutions are discussed. Furthermore, rogue wave solutions are derived through the degeneration of breather-positon solutions.

Extension of an Eighth-Order Iterative Technique to Address Non-Linear Problems

The convergence order of an iterative method used to solve equations is usually determined by using Taylor series expansions, which in turn require high-order derivatives, which are not necessarily present in the method. Therefore, such convergence analysis cannot guarantee the theoretical convergence of the method to a solution if these derivatives do not exist. However, the method can converge. This indicates that the most sufficient convergence conditions required by the Taylor approach can be replaced by weaker ones. Other drawbacks exist, such as information on the isolation of simple solutions or the number of iterations that must be performed to achieve the desired error tolerance. This paper positively addresses all these issues by considering a technique that uses only the operators on the method and Ω-generalized continuity to control the derivative. Moreover, both local and semi-local convergence analyses are presented for Banach space-valued operators. The technique can be used to extend the applicability of other methods along the same lines. A large number of concrete examples are shown in which the convergence conditions are fulfilled.

Incremental Nonlinear Dynamics Inversion Control with Nonlinear Disturbance Observer Augmentation for Flight Dynamics

A flight controller formulation based on incremental nonlinear dynamics inversion (INDI) control with nonlinear disturbance observer (NDO) is proposed. INDI control is a nonlinear controller based on incremental dynamics. Aimed to attain robustness for nonlinear dynamics inversion (NDI)-based controller, incremental dynamics are derived using the first-order Talyor series expansion to nonlinear systems. The incremental dynamics-based controller requires information on state derivative terms to strengthen the robustness property of the nonlinear controller. The proposed controller utilizes the first-order low-pass filter to obtain the state derivative estimate to implement incremental dynamics into the system. Because the incremental form creates uncertainty term which is an aftermath of the Taylor series expansion, the proposed controller adopts the NDO to eliminate this effect. The controller is applied to the generic transport model which was developed by NASA for simulation purposes. The proposed NDO-based INDI control underwent simulations, together with an INDI controller without disturbance observer, and showed that the developed method results in better performances, providing important advantages where it compensates the uncertainties, removes the steady-state error, and shows less oscillating longitudinal body rate response than the baseline controller, desirable for aerodynamics applications with faster system response.

Euclidean direction search algorithm with maximum correntropy criterion for active noise control system

Based on the principle of superposition, the active noise control (ANC) technique can achieve satisfactory noise reduction. The filtered-x least mean square (FxLMS) algorithm has been extensively implemented in the ANC problem but it is easily plunged into instability in impulsive noise scenarios. To ameliorate this disadvantage, benefiting from the reduced computational load of the Euclidean direction search (EDS) algorithm and robustness of the maximum correntropy criterion (MCC), a novel filtered-x EDS-MCC (FxEDS-MCC) algorithm is proposed to attenuate the impulsive interference. The theoretical analysis of the FxEDS-MCC algorithm is based on the rotated method and the Taylor series expansion approximation. Simulations validate the accuracy of the theoretical performance and verify the improved performance of the FxEDS-MCC algorithm in comparison with the existing algorithms.

Strong CP problem in the quantum rotor

Recent studies have claimed that the strong CP problem does not occur in QCD, proposing a new order of limits in volume and topological sectors when studying observables on the lattice. In order to shed light on this issue, we study the effect of the topological θ-term on a simple quantum mechanical rotor that allows a lattice description. The topological susceptibility and the θ-dependence of the energy spectrum are both computed using local lattice correlation functions. The sign problem is overcome by considering Taylor expansions in θ, exploiting automatic differentiation methods for Monte Carlo processes. Our findings confirm the conventional wisdom on the strong CP problem. Published by the American Physical Society 2024

Compact Analysis of the Necessity of Padé Approximation for Delayed Continuous-time Models in LQR, H-Infinity and Root Locus Control Strategies

This paper presents a comprehensive analysis of the need for the Padé approximation for continuous-time models with delays, focusing on its critical role in addressing the control challenges posed by time delays. Time delays, often referred to as dead times, transport delays or time lags, are inherent in a wide range of industrial and engineering processes. These delays introduce phase shifts that degrade control performance by reducing control bandwidth and threatening the stability of closed-loop systems. Accurate modelling and compensation of these delays is essential to maintain system stability and ensure effective control. This paper highlights the difficulties that arise when using advanced control techniques such as root locus (RL), linear quadratic regulator (LQR) and H-infinity (H_∞) control in systems with delays. Representing delays in exponential form leads to an infinite number of state problems, complicating the design and analysis of controllers in such systems. To address these challenges, the Padé approximation is proposed as an effective method for approximating time delays with rational polynomials of appropriate order. This approach allows for more accurate simulation, system analysis and controller design, thereby mitigating the problems caused by delays. The paper also provides a detailed comparative analysis between the Padé approximation and Taylor polynomials, demonstrating the superiority of the former in achieving accurate delay modelling and control performance. The results show that the use of Padé approximation not only improves the accuracy of system models, but also improves the robustness and stability of control strategies such as RL, LQR, and H_∞. These results highlight the importance of the Padé approximation as a valuable tool in the design of delay-affected control systems, offering significant advantages for both theoretical and practical applications.

Theoretical analysis and predictions for the two-neutrino double electron capture of 124Xe

Abstract We provide a complete theoretical description of the two-neutrino electron capture in 124Xe, improving both the nuclear and the atomic structure calculations. We improve the general formalism through the use of the Taylor expansion method, leading to higher-order terms in the decay rate of the process. The nuclear part is treated with pn-QRPA and interacting shell model (ISM) methods. The nuclear matrix elements (NMEs) are calculated with the pn-QRPA method with isospin restoration by fixing the input parameters so that the experimental decay rate is reproduced, resulting in values significantly lower than in previous calculations. The validity of the pn-QRPA NMEs is tested by showing their values to be comparable with the ones for double-beta decay with the emission of two antineutrinos of 128,130Te, which have similar pairing features. Within the ISM, we reproduce the total experimental half-life within a factor of two and predict the capture fraction to the KK channel of about 74%. We also predict the capture fractions to other decay channels and show that for the cumulative decay to the KL1–KO1 channels, a capture fraction of about 24% could be observed experimentally. On the atomic side, calculations are improved by accounting for the Pauli blocking of the decay of innermost nucleon states and by considering all s-wave electrons available for capture, expanding beyond the K and L1 orbitals considered in previous studies. We also provide improved atomic relaxation energies of the final atomic states of 124Te, which may be used as input for background modeling in liquid Xenon experiments.

Study on Image Classification Algorithm Based on Multi-Scale Feature Fusion and Domain Adaptation

This paper introduces the MMTADAN, an innovative algorithm designed to enhance cross-domain image classification. By integrating multi-scale feature extraction with Taylor series-based detail enhancement and adversarial domain adaptation, the MMTADAN effectively aligns features between the source and target domains. The proposed approach addresses the critical challenge of generalizing classification models across diverse datasets, demonstrating significant improvements in performance. The findings suggest that retaining essential image details through multi-scale extraction and Taylor series enhancement can lead to better classification outcomes, making the MMTADAN a valuable contribution to the field of image classification.

Modelling the formation kinetics of Fe2B layers with a diffusionmodel using taylor expansion

Modelling the formation kinetics of Fe2B layers with a diffusionmodel using taylor expansion

Ramanujan-Berndt-type integrals of order four and series associated with Jacobi elliptic functions

In this paper, we evaluate in closed forms two families of infinite integrals containing hyperbolic and trigonometric cosine functions in their integrands. We call them Ramanujan-Berndt-type integrals of order four since they initiated the study of similar integrals. We establish explicit evaluations of twelve classes of hyperbolic sums by special values of the Gamma function by two completely different approaches, which extend those sums considered by Xu, Yang, Zhao and the second author previously. We discover the first by refining two results of Ramanujan concerning some q-series. For the second we compare both the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions. Furthermore, by contour integrations we convert two families of Ramanujan-Berndt-type integrals of order four to the above hyperbolic sums, all of which can be evaluated in closed forms. We then discover explicit formulas for one of the two families. Moreover, we provide many examples which enable us to formulate a conjectural explicit formula for the other family of the Ramanujan-Berndt-type integrals at the end, and obtain some results on Barnes' multiple zeta function.