Taylor Series Research Articles

A Monte Carlo Investigation of Confidence Intervals for a Nondecreasing Series

We consider the problem of estimating a time series of population means from a series of sample surveys when the means are known to be nondecreasing. We introduce the standard survey estimators of the series of means, which are not guaranteed to be nondecreasing. We employ the Pool Adjacent Violators Algorithm (PAVA) to turn the series of standard survey estimates into a nondecreasing series. We introduce five methods of constructing confidence intervals for the series of non-decreasing means: normal-theory intervals based on the standard survey point estimator of the mean and the Taylor series estimator of its variance; normal-theory intervals based on the nondecreasing PAVA estimator of the mean and a unique jackknife estimator of its variance developed here (jackknife-[Formula: see text]); intervals similar to the aforementioned method but based on Student’s-[Formula: see text] distribution (jackknife-[Formula: see text]); analytical intervals due to Morris; and simultaneous confidence limits due to Korn. We report the results of a Monte Carlo simulation and assess the methods’ performance under various scenarios. Jackknife-[Formula: see text] intervals exhibited coverage probabilities that are near the nominal value. Taylor, jackknife-[Formula: see text], and Korn intervals found coverage below the nominal value while Morris intervals were excessively conservative. Jackknife intervals were much narrower than Taylor intervals.

CSKINet: A multimodal network model integrating conceptual semantic knowledge injection for relation extraction of Chinese corporate reports

Recognizing the associations among entities in corporate reports accurately is crucial for market regulation and policy development. Nevertheless, confronted with massive corporate information, the traditional manual screening approach is cumbersome, struggling to match the demand. Consequently, we propose a multimodal network model incorporating conceptual semantic knowledge injection, CSKINet, for accurately extracting relations from Chinese corporate reports. The essential highlights in the design of the CSKINet model are the following: (1) Integrate the conceptual descriptions of corporations from external resources to construct the semantic knowledge repository of corporate concepts, which provides a solid semantic foundation for the model. (2) Multimodal features are extracted from the documents by various means and corporate conceptual knowledge is integrated into the model representation to enhance the representation capability of the model. (3) The multimodal self-attention mechanism that captures cross-modal associations and the biaffine classifier with Taylor polynomial loss function that optimizes training iterations further improve the learning efficiency and prediction accuracy. The results on the real corporate report dataset show that our proposed model can more accurately extract the relations from Chinese corporate reports compared to other baseline models, where the F1 score reaches 85.76%.

An Integrated Taylor Expansion and Least Squares Approach to Enhanced Acoustic Wave Staggered Grid Finite Difference Modeling

The staggered grid finite difference method has emerged as one of the most commonly used approaches in finite difference methodologies due to its high computational accuracy and stability. Inevitably, discretizing over time and space domains in finite difference methods leads to numerical artifacts. This paper introduces a novel approach that combines the widely used Taylor series expansion with the least squares method to effectively suppress numerical dispersion. We have derived the coefficients for the staggered grid finite difference method by integrating Taylor series expansions with the least squares method. To validate the effectiveness of our approach, we conducted analyses on accuracy, dispersion, and stability, alongside simple and complex numerical examples. The results indicate that our method not only inherits the capabilities of the original Taylor series and least squares methods in suppressing numerical dispersion across small and medium wavenumber ranges but also surpasses the original methods. Moreover, it demonstrates robust dispersion suppression capabilities at high wavenumber ranges.

Uncertainty quantification in the modal analysis of aircraft stiffeners: A Perturbation Technique approach in SFEM

Structural members, such as stiffeners, are crucial in aeronautical structures, providing essential dynamic stiffness. However, variability introduced by manufacturing and assembly processes, as well as material inconsistencies, can lead to uncertainties in structural performance. It is crucial to incorporate these uncertainties into structural analysis to ensure reliable design. This paper utilizes the Stochastic Finite Element Method (SFEM) to address uncertainties in typical structural members used in aeronautics. The Perturbation Technique, based on Taylor series expansions, was employed to model uncertainties in aircraft stiffeners. The study focuses on natural frequencies and modal analysis to evaluate the impact of uncertainties on beams with hat and Z sections, commonly used as stiffeners in aircraft panels. These stiffeners were modeled using the Timoshenko beam theory, and sensitivity analysis was performed to identify key contributors to variability. The perturbation parameter was validated through Monte Carlo simulations. Sensitivity analysis, employing gradient-based methods, identified significant factors affecting variability in natural frequencies. A different perturbation parameter was necessary based on the stiffener's geometry: thickness variations required a perturbation parameter on the order of 10−3, whereas dimensions changes in the flange and height required parameters on the order of 10−2. These results underscore the importance of choosing appropriate perturbation magnitudes to avoid inaccuracies in the deterministic frequency response. Once a perturbation parameter is established, it can be applied to similar regions, ensuring the robustness of the SFEM methodology in analyzing the dynamic response of aeronautical structural reinforcements.

Variance estimation using memory type estimators based on EWMA statistic for time scaled surveys in stratified sampling

In this article, we have proposed memory-type exponential and non-exponential estimators for population variance based on exponentially weighted moving average (EWMA) statistic in stratified sampling. We drive mathematical expressions for mean square errors of the proposed estimators using Taylor and exponential expansions. Our analysis demonstrates that the proposed memory-type estimators outperform the conventional estimators under stratification, particularly, when the information of previous sample is utilized. The performances of the proposed estimators are evaluated mathematically by deriving the conditions in which the memory-type estimators would perform better than their corresponding conventional estimators under stratification. Through an extensive simulation, we evaluated the performance of the proposed estimators across various population parameters, revealing their enhanced efficiency in time-scaled survey. Additionally, a real data application is also used to support the mathematical findings, confirming the practical utility of the proposed estimators. The results of numerical study underscore the importance of the use of previous sampled information which significantly improves the accuracy and reliability of the proposed estimator for variance estimation for time-scaled surveys.

Expansion of Landau theory with Magnetic Susceptibility and Heat Capacity

Abstract. Landau theory employs free energy to depict the work potential of molecules and utilizes order parameters to signify the degree of molecular organization. By applying a Taylor expansion of the free energy with respect to the order parameter in the vicinity of a phase transition, the theory elucidates how molecular arrangements can influence the system's energy. The universality of Landau theory stems from the order parameter within the framework of free energy expansion. The application of critical exponents exemplifies the universality of Landau theory, showcasing the properties of materials that share the same dimensionality and similar correlation lengths. This article aims to derive formulas and discuss practical applications of Landau theory, such as analyzing the heat capacity and magnetic susceptibility of materials across a range of temperatures.

A fast calculation method for three-impulse contingency return trajectory during the near-moon phase based on differential algebra

Aimed at the demand of contingency return at any time during the near-moon phase in the manned lunar landing missions, a fast calculation method for three-impulse contingency return trajectories is proposed. Firstly, a three-impulse contingency return trajectory scheme is presented by combining the Lambert transfer and maneuver at the special point. Secondly, a calculation model of three-impulse contingency return trajectories is established. Then, fast calculation methods are proposed by adopting the high-order Taylor expansion of differential algebra in the two-body trajectory dynamics model and perturbed trajectory dynamics model. Finally, the performance of the proposed methods is verified by numerical simulation. The results indicate that the fast calculation method of two-body trajectory has higher calculation efficiency compared to the semi-analytical calculation method under a certain accuracy condition. Due to its high efficiency, the characteristics of the three-impulse contingency return trajectories under different contingency scenarios are further analyzed expeditiously. These findings can be used for the design of contingency return trajectories in future manned lunar landing missions.

Consistent theories for the DESI dark energy fit

We search for physically consistent realizations of evolving dark energy suggested by the cosmological fit of DESI, Planck and Supernovae data. First we note that any lagrangian description of the standard Chevallier-Polarski-Linder (CPL) parametrization for the dark energy equation of state w, allows for the addition of a cosmological constant. We perform the cosmological fit finding new regions of parameter space that however continue to favour dark energy with w < -1 at early times, that is challenging to realize in consistent theories. Next, in the spirit of effective field theories, we consider the effect of higher order terms in the Taylor expansion of the equation of state of dark energy around the present epoch. We find that non-linear corrections of the equation of state are weakly constrained, thus opening the way to scenarios that differ from CPL at early times, possibly with w > -1 at all times. We present indeed scenarios where evolving dark energy can be realized through quintessence models. We introduce in particular the ramp model where dark energy coincides with CPL at late times and approximates to a cosmological constant at early times. The latter model provides a much better fit than ΛCDM, and only slightly worse than w 0 w a CDM, but with the notable advantage of being described by a simple and theoretically consistent lagrangian of a canonical quintessence model.

HIV Outcomes Among Women Living With HIV Who Experienced Early Sexual Violence Across Four Sub-Saharan African Countries.

Early experiences of sexual violence may influence HIV care and treatment outcomes among women living with HIV (WLHIV). We examined whether self-report by WLHIV of being forced into their first sexual experience was associated with awareness of HIV-positive status, being on antiretroviral therapy (ART) and being virologically suppressed. We conducted a secondary analysis using nationally representative, cross-sectional Population-based HIV Impact Assessment surveys from Lesotho, Malawi, Zambia, and Zimbabwe conducted from 2015 through 2017. Adjusted logistic regression models with survey weights and Taylor series linearization were used to measure the association between forced first sex and 3 HIV outcomes: (1) knowledge of HIV status among all WLHIV, (2) being on ART among WLHIV with known status, and (3) virological suppression among WLHIV on ART. Among WLHIV, 13.9% reported forced first sex. Odds of knowledge of HIV status were not different for WLHIV with forced first sex compared with those without (adjusted odds ratio [aOR], 1.17; 95% CI: 0.95 to 1.45). Women living with HIV with forced first sex had significantly lower odds of being on ART (aOR 0.74, 95% CI: 0.57 to 0.96) but did not have lower odds of virological suppression (aOR 1.06, 95% CI: 0.80 to 1.42) compared with WLHIV without forced first sex. While high proportions of WLHIV were on ART, report of nonconsensual first sex was associated with a lower likelihood of being on ART which may suggest that early life trauma could influence long-term health outcomes.

A Taylor Series Approximation Model for Characterizing the Output Resistance of a GFET

A Taylor Series Approximation Model for Characterizing the Output Resistance of a GFET

The Non-Perturbative Approach in Examining the Motion of a Simple Pendulum Associated with a Rolling Wheel with a Time-Delay

The present study aims to examine the movement of a simple pendulum (SP) that is connected by a lightweight spring and connected with a rotating wheel. The motivation behind this topic is to gain a comprehensive understanding of intricate dynamic systems that involve interconnected mechanical components and feedback with temporal delays. This system is not only theoretically attractive but also practically applicable in domains such as robotics, engineering, and control systems. As well known, all classical perturbation methods exploit Taylor expansion to simplify the reality of restoring forces. In contrast, the non-perturbative approach (NPA), as a novel methodology, transforms any nonlinear ordinary differential equation (ODE) into a linear one. It scrutinizes the restoring forces, away from employing Taylor expansion; hence it eliminates the previous weakness. The concept of the NPA is based mainly on the He’s frequency formula (HFF). The confidence of the NPA comes from the numerical compatibility between the nonlinear and linear ODEs via the Mathematica Software (MS). Therefore, instead of handling the nonlinear ODE, we investigate the linear one. The achieved response is plotted over time to show the impact of the acted parameters during a specified time interval. Moreover, the phase plane curves that correspond to the plotted solution are presented and examined. The stability criteria of the analogous linear ODE are provided and drawn to explore the stability/instability zones. The performance is applicable in engineering and other fields due to its ease of adaptation to different nonlinear systems. Therefore, the NPA can be regarded as substantial, successful, and interesting and can extended to be applied in further categories within the field of couples dynamical systems.

Comparative analysis of accelerated gradient algorithms for convex optimization: high and super resolution ODE approach

We investigate convex differentiable optimization and explore the temporal discretization of damped inertial dynamics driven by the gradient of the objective function. This leads to three accelerated gradient algorithms: Nesterov Accelerated Gradient (NAG), Ravine Accelerated Gradient (RAG), and (IGAHD). The latter was introduced by discretizing inertial dynamics with Hessian-driven damping to attenuate inherent oscillations in inertial methods. By analysing the high-resolution ODEs of order p = 0,1,2 for these algorithms, we gain insights into their similarities and differences. All three algorithms share the same low-resolution ODE of order 0, which is the dynamic proposed as a continuous surrogate for (NAG). To differentiate Nesterov from Ravine, we refine the comparison and demonstrate distinct high-resolution ODEs of order 2 in h (termed super-resolution). The corresponding Taylor expansions in h reveal matching terms of order 1 but differing terms of order 2. To the best of our knowledge, this result is completely new and emphasizes the need to avoid confusion between the Ravine and Nesterov methods in the literature. We present numerical experiments to illustrate our theoretical results. Performance profiles, measuring the number of iterations, indicate that (IGAHD) outperforms both (NAG) and (RAG) methods. (RAG) exhibits a slight advantage over (NAG) in terms of the average number of iterations. When considering CPU-time, both (RAG) and (NAG) outperform (IGAHD). All three algorithms exhibit similar behaviour when evaluating based on gradient norms.

On Efficient Method For Fractional-Order Two-Dimensional Navier-Stokes Equations

For the solution of fractional-order two-dimensional Navier-Stokes equations (FOTDNSEs), the current work proposes a novel variant of the Laplace Adomian decomposition approach with the Atangana-Baleanu fractional operator in the Caputo sense (ABCFO). In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the exact solution. Only two components are required for the new approximation analytical technique. When compared to other strategies, the current method is very simple, requires less calculation, and is extremely accurate.

Invariance Principle and Berry–Esseen Bound for Error Variance Estimator for Pth-Order Nonlinear Autoregressive Models

The invariance principle and Berry–Esseen bound for an error variance estimator based on the residuals are established by using a Taylor expansion and the classical invariance principle and Berry–Esseen bound for independent random variables. Some examples are given to illustrate their applications.

Applications of Taylor Series in Approximate Calculation and Limits

Taylor series plays a significant role in mathematics, especially in the fields of mathematical analysis, calculus and applied mathematics. It is not only a theoretical tool, but also an indispensable method in practical applications. For instance, it plays an important role in the fields of physics, engineering and economics. Taylor function uses function which is represented by a series of infinite additive terms, which are calculated by the derivative of the function at a certain point. This paper mainly uses literature research method and empirical research method to introduce the basic information and practical application of Taylor series. Some methods are given to solve related problems. This article demonstrates a brief introduction of Taylor series, then illustrates the steps of proof for Taylor series and provides the solutions of several practical examples. In the end, the paper focuses on the application of Taylor series on limits and analyzes some typical and interesting examples.

Stochastic Lighthill-Whitham-Richards traffic flow model for nonlinear speed-density relationships

Stochasticity is becoming increasingly essential in traffic flow research, given its notable influence in several applications, such as real-time traffic management. To consider stochasticity in macroscopic traffic flow modeling, this paper introduces a stochastic Lighthill-Whitham-Richards (SLWR) model, which not only captures equilibrium values in steady-state conditions but also describes stochastic variabilities. The SLWR model follows a conservation law, in which the free-flow speed is randomized to represent heterogeneities of drivers. To more accurately reflect real-life traffic patterns, a nonlinear speed-density relationship is considered. For addressing this highly nonlinear problem, a dynamically bi-orthogonal (DyBO) method is coupled with the Taylor series expansion technique. The results of simulation experiments show that the SLWR model can effectively describe the evolution of stochastic dynamic traffic with temporal or geometric bottlenecks. Moreover, the DyBO solutions exhibit reasonable accuracy while significantly reducing computation costs compared with the Monte Carlo method.

Taylor's formula on the separating complex plane and its applications

This article begins by exploring split-complex numbers, also known as dual or double numbers, a mathematical concept that extends the real number system by creating a commutative ring with a zero divisor. Furthermore, the paper extends the Taylor series theory from the real analysis domain to the domain of split-complex numbers, thereby establishing the corresponding Taylor formula on the split-complex plane. The introduction of split-complex numbers opens up new perspectives and provides new tools for the modeling and analysis of quantum systems. In the field of electromagnetism, the application of split-complex numbers also demonstrates its potential, contributing to a more precise description of the behavior of electromagnetic fields under specific conditions. These theoretical advancements are expected to further promote research in the field of split-complex analysis, playing a more critical role in important branches of physics such as quantum mechanics and electromagnetism, thereby facilitating the development of related theories and the emergence of innovative practical applications.

A conformable fractional finite difference method for modified mathematical modeling of SAR-CoV-2 (COVID-19) disease.

In this research, the ongoing COVID-19 disease by considering the vaccination strategies into mathematical models is discussed. A modified and comprehensive mathematical model that captures the complex relationships between various population compartments, including susceptible (Sα), exposed (Eα), infected (Uα), quarantined (Qα), vaccinated (Vα), and recovered (Rα) individuals. Using conformable derivatives, a system of equations that precisely captures the complex interconnections inside the COVID-19 transmission. The basic reproduction number (R0), which is an essential indicator of disease transmission, is the subject of investigation calculating using the next-generation matrix approach. We also compute the R0 sensitivity indices, which offer important information about the relative influence of various factors on the overall dynamics. Local stability and global stability of R0 have been proved at a disease-free equilibrium point. By designing the finite difference approach of the conformable fractional derivative using the Taylor series. The present methodology provides us highly accurate convergence of the obtained solution. Present research fills research addresses the understanding gap between conceptual frameworks and real-world implementations, demonstrating the vaccination therapy's significant possibilities in the struggle against the COVID-19 pandemic.

Portfolio Selection with Hierarchical Isomorphic Risk Aversion

Researchers usually specify risk aversion coefficients from 1 (lowest) to M (highest) for a portfolio to indicate active or passive approaches. How effective is this practice? Recent studies suggest that the global minimum variance portfolio (GMVP) is statistically equivalent to portfolios with extensive risk aversion coefficients (the GMVP-equivalent). Expressing the risk aversion coefficient as a Taylor series of the target return and efficient set constants, we generalize the previous result to the non-GMVP-equivalents and segment mean-variance portfolios according to a hierarchy of risk aversion coefficients. In this paper, we show that hierarchical risk aversion coefficients are superior to isometric attributes.

On λ-Pseudo Bi-Starlike Functions Related to Second Einstein Function

A new class BΣλ(γ,κ) of bi-starlike λ-pseudo functions related to the second Einstein function is presented in this paper. c2 and c3 indicate the initial Taylor coefficients of ϕ∈BΣλ(γ,κ), and the bounds for |c2| and |c3| are obtained. Additionally, for ϕ∈BΣλ(γ,κ), we calculate the Fekete–Szegö functional.