Search Results (463)

The convergence order of higher-order iterative methods for solving systems of nonlinear equations was analyzed using Taylor series expansion, which typically requires the computation of higher-order derivatives not inherently part of the method. This dependency limits the method’s applicability and increases the computational cost. The distinctiveness of our work lies in the development of improved convergence theorems that rely solely on first-order derivatives. The proposed approach offers a stronger framework than existing methods by incorporating details about the convergence region’s radius and providing precise error estimates. Furthermore, we explore semi-local convergence, which holds greater significance as it allows the identification of the specific domain where the iterative sequence remains valid. The theoretical findings are substantiated through suitable numerical illustrations. Full article

In this paper, we propose memory-type ratio, product, exponential ratio, and exponential product estimators based on an exponentially weighted moving average (EWMA) statistic for mean estimation in time-scaled surveys using systematic sampling. The approximate expressions for bias and mean squared error (MSE) of the proposed memory-type estimators are derived using Taylor and exponential series expansions up to the second order. Mathematical conditions are also established under which the memory-type ratio, product, exponential ratio, and exponential product estimators outperform their corresponding conventional estimators in terms of MSE. A simulation study is carried out to evaluate the performance of the proposed estimators compared to the conventional estimators. Additionally, real data application is also used to support the simulation findings. The results indicate that the proposed memory-type estimators have smaller MSEs than conventional estimators for time-scaled surveys using systematic sampling. Full article

The Hyakuzahōdan Kikigakishō (百座法談聞書抄, hereafter Hyakuza 百座), compiled in the late Heian period, is an important Buddhist document that records a hundred-day lecture series on the Lotus Sutra (法華経). While previous scholarship has recognized the constructed nature of the text as a kikigaki (聞書), it has predominantly focused on content analysis, implicitly treating the text as a transparent window into the actual preaching event. To move beyond this limitation, this study proposes the analytical framework of dual performativity and, drawing on Diana Taylor’s theory of the archive and the repertoire, reexamines the text’s generative logic and political implications. This study argues that the Hyakuza embodies two interrelated forms of performance: first, the performativity of the hōdan (法談) as a live ritual, understood as a repertoire performance that constructs immediate authority through body, voice, and situational dynamics; second, the performativity of the kikigaki as textual construction, understood as an archival performance that transforms the ephemeral oral event into an authoritative, transmissible text through formulaic rhetoric, localized adaptation, and systematic arrangement. Integrating methodologies from textual history, rhetorical analysis, ritual theory, and intellectual history, this study demonstrates that the Hyakuza is not a neutral transcript of sermons but a meticulous, intentional act of writing with two fundamental aims: on a cultural level, to hierarchically integrate shinbutsu shūgō (神仏習合) through narrative appropriation; on a social level, to symbolically bind Buddhist merit with the institutional identities of aristocrats such as naishinnō (内親王), ultimately serving the self-affirmation internal cohesion, and cultural demarcation of the elite community from the masses, while simultaneously contributing to the state’s project of constructing a unified ideology in the late Heian period. By examining both cross-civilizational universal logic and specific historical context, this study reveals how the Hyakuza’s dual performativity produces and categorizes knowledge narratives while embedding political power dynamics, offering a critical path for the study of kikigaki-genre literature from discourse analysis to politics of textuality. Full article

This paper introduces a new explicit integration method for second-order ordinary differential equations (ODEs) commonly encountered in engineering applications. Traditionally, these problems are solved either by reformulating them as first-order systems to apply one-step methods such as Runge–Kutta schemes, or by using direct second-order approaches widely adopted in linear dynamics, including the generalized-α, central difference, and Newmark methods. The proposed method is derived from a Taylor series expansion truncated at the third derivative, resulting in a fully explicit algorithm that requires only one function evaluation per time step. Similar to Newmark’s formulation, it includes adjustable parameters that allow the user to balance accuracy and stability. For a specific parameter choice, the method exhibits convergence and stability properties comparable to those of the central difference scheme. An important advantage is that it remains explicit even when nonlinearities depend on first-derivative terms. The paper presents a theoretical analysis covering stability, local truncation error, spectral properties, numerical damping, and period elongation. The method is validated through four test cases from multibody dynamics, including linear and nonlinear problems. Results demonstrate that the Explicit Integration Grade 3 (EIG-3) method achieves accuracy comparable to existing explicit second-order integrators while significantly reducing computational cost, particularly in nonlinear applications. Full article

This paper presents a comprehensive review of digital floating-point arithmetic algorithms that utilize Taylor series expansion in combination with mantissa-region division techniques, and it further demonstrates their generalization and applicability based on the findings of our research. While the discussion is broad in scope, this paper consolidates and systematizes the authors’ method within a broader contextual discussion, rather than presenting a fully systematic review of the entire state of the art in floating-point arithmetic algorithms. In many scientific computing applications, compact and low-power hardware implementations are essential. To address these requirements, this review presents algorithms specifically designed to operate under such constraints. The focus is placed on efficient floating-point operations—including division, inverse square root, square root, exponentiation, and logarithmic functions—all realized through Taylor series expansion with mantissa region division techniques. Furthermore, the trade-offs are examined in detail, covering factors such as the required numbers of additions, subtractions, and multiplications, along with the look-up table (LUT) size. The study further identifies the environments and application domains where the Taylor series expansion method combined with mantissa-region division is most effective, based on comparisons with various other floating-point computation algorithms and their corresponding hardware implementations. Overall, the review underscores the value of this unified framework in enabling efficient and adaptable floating-point computation across a wide range of hardware-constrained environments. Full article

Time series forecasting in power systems is crucial for power supply planning and exerts a direct impact on the electricity market. Accurate forecasting can effectively mitigate decision-making risks. This paper proposes a forecasting method based on a multi-dimensional Taylor network (MTN) and applies it to electricity price prediction. The time series is decomposed into one low-frequency signal and several high-frequency signals. The MTN model is constructed for each frequency sequence. The final forecast is obtained by aggregating the predictions from all frequency components. Using European electricity price data as a case study, experimental results demonstrate that the proposed method achieves high predictive accuracy. Full article

The existing methods for solving non-linear equations encounter convergence issues and computing constraints, especially when used in fractional-order or complex non-linear problems. This study develops a higher-order fractional technique for solving non-linear equations based on the Caputo fractional derivative. The proposed method uses a fractional framework to improve local convergence and stability while ensuring high efficiency in every iteration step. Local convergence analysis using generalized Taylor series expansion reveals that the order of the new fractional scheme for solving non-linear equations is $5¢+1$, where $¢\in$ (0,1] represents the Caputo fractional order, determining the memory depth of the Caputo fractional derivative. The performance of the method is further investigated using a variety of non-linear problems from engineering optimization and applied sciences, such as engineering control systems, computational chemistry, thermodynamics models, and operations research, such as inventory optimization. Analyzing the key performance metrics, such as dynamical analysis, percentage convergence, residual error, and computation time, confirms the advantages of the developed method over the state-of-the-art. This study provides a solid framework for higher-order fractional iterative approaches, paving the way for advanced applications of non-linear problems. Full article

This study focuses on enhancing wind speed prediction for wildfire spread simulation by proposing an integrated forecasting approach. The original wind speed series is first processed via variational mode decomposition (VMD), with its parameters [K, $\alpha$] optimized via particle swarm optimization (PSO). Every intrinsic mode function (IMF) resulting from this decomposition is predicted using a bidirectional long short-term memory model incorporating an attention mechanism (AM-BiLSTM), and the final wind series is reconstructed from these predictions. Model training and validation were conducted using data from controlled burning experiments in the Mao’er Mountain area of Heilongjiang Province, China. Predictive performance is evaluated through multiple statistical metrics, error distribution analysis, and Taylor diagrams. To assess practical utility, the predicted wind field is further applied in FARSITE to drive wildfire spread simulations. Results demonstrate that the PSO-VMD-AM-BiLSTM model provides reliable wind forecasts and contributes to improved fire spread prediction accuracy, indicating its potential for decision support in wildfire management. To achieve accurate forest fire spread prediction, we construct the MCNN model, which is based on early perception of understory wind fields using predicted wind speed data and adopts a multi-branch convolutional neural network architecture to extract fire spread features. FARSITE is employed to simulate forest fire spread in the Mao’er Mountain region, generating a dataset for model training and testing. After 50 training epochs, the loss value of the MCNN model converges, achieving optimal prediction performance when the combustion threshold is set to 0.7. Compared to models such as CNN, DCIGN, and DNN, MCNN shows improvements in evaluation metrics including precision, recall, Sørensen coefficient, and Kappa coefficient. To validate the model’s predictive performance in real fire scenarios, four field ignition experiments were conducted at the Liutiao Village test site: homogeneous fuel combustion, long fire line combustion, alternating fuel combustion, and multiple ignition source merging combustion. Comprehensive evaluation across the four experiments indicates that the model achieves precision, recall, Sørensen coefficient, and Kappa coefficient values of 0.940, 0.965, 0.953, and 0.940, respectively, with stable prediction errors below 6%. These results represent improvements over the comparative models DCIGN and DNN. The proposed MCNN model can adapt to forest fire spread prediction under different scenarios, offering a novel approach for accurate forest fire prediction and prevention. Full article

The Taylor approximation theorem is a fundamental tool in numerical analysis, providing a local polynomial representation of smooth functions. In practical computations, a function f is approximated by a finite Taylor polynomial $P_n$, and controlling the resulting truncation error is of central importance. In this paper, we introduce two novel a posteriori error estimation techniques for Taylor polynomial approximations. The proposed estimators are fully computable and do not require prior bounds on the $(n+1)$st derivatives of f. We prove that the estimators converge to the exact error both pointwise and in the $L^2$-norm as $n\to \infty$, and we establish their asymptotic sharpness through effectivity analysis. Based on these results, we develop two adaptive algorithms that automatically determine the minimal degree n required to achieve a prescribed tolerance, either at a specific point or over a domain. We further extend the analysis to multivariate functions and show that analogous estimators and effectivity properties hold in higher dimensions. Numerical experiments are presented to validate the theoretical results and demonstrate the practical performance of the proposed methods. Full article

The applicability of a highly efficient sixth-order convergent method, originally proposed by Kansal et al., is extended in this study to a Banach space setting. The initial development of this method relied upon Taylor series expansions in ${\mathbb{R}}^n$ and the assumption that the nonlinear operator is sufficiently differentiable. This vague condition implies the existence of high-order derivatives that are not actually utilized by the algorithm. This study transcends these limitations by establishing convergence based solely on generalized continuity conditions of the first Fréchet derivative. By dispensing with these strong smoothness requirements, the domain of applicability is significantly widened. We derive computable radii for the ball of convergence and establish error bounds under local analysis. Furthermore, a rigorous semi-local convergence analysis is presented, a feature previously absent in the literature for this specific scheme, utilizing a majorizing sequence technique to guarantee the existence and uniqueness of the solution. The theoretical results are validated through numerical experiments, which demonstrate that the method converges even when the standard sufficiently differentiable conditions are violated. Full article

Taylor–Young and Maclaurin series are widely used for approximating smooth functions around a given point. This study investigates a unified stochastic framework for Taylor-type expansions of functions by means of independent random variables. The proposed probabilistic expansions of a function are able to incorporate evaluations of derivatives at different points, leading to a global approach. Exact expansions are obtained for any order of available derivatives, and such Taylor-type expansions enable the statistical inference of the remainder terms. It appears that the traditional Taylor–Young and Maclaurin series are particular cases of the proposed approach thanks to the Dirac probability measure, and guidelines for using Taylor series have been enhanced. Different ways of choosing the optimal distributions of random variables are provided, particularly when truncations are applied. Numerical comparisons are provided as well. Full article

The use of integral and differential operators in geometric function theory has continued to gain interest among researchers in the field of study in recent times. This is due to the wide range of its applications in science, technology and engineering. In this work, therefore, the authors defined and investigated a new subclass of analytic functions in the open unit disk using the $\mathfrak{q}$-Srivastava–Attiya convolution operator and the Jackson’s $\mathfrak{q}$-derivative, by means of the subordination. The authors used two well-known lemmas to determine a sharp upper-bound for the Fekete–Szeg$\ddot{o}$ functional in two different cases. In particular, the authors introduced a new generalized subclass of complex order univalent functions denoted by ${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,\Phi \right)$ and derived the coefficient estimates $a_{\iota }(\iota =2,3)$ of the Taylor–Maclaurin series in this class, as well as the Fekete–Szeg$\ddot{o}$ inequality $\left|a_3-\aleph a_2^2\right|$ for functions in this class. The work generalizes many known results in the literature. Full article

In this article, the problem of control of the kinematic model of an omnidirectional robot with time delay in the control input is tackled through an Active Disturbance Rejection Control (ADRC) with a disturbance predictor-based scheme, which consists in predicting the generalized forward disturbance input in order to cancel it and then using a feedforward linearization approach to control the system in trajectory tracking tasks. The novelties of the scheme are to demonstrate that using the proposed extended state disturbance estimation leads to a forward estimation following the Taylor series approximation, and, to avoid using additional pose predictions, a feedforward input as an exact linearization approach is used, in which the remaining dynamics can be lumped into the generalized disturbance input. Thus, the use of extended states in prediction improves the robustness of the predictor while increasing the prediction horizon for larger time delays. The stability of the proposal is demonstrated using the second method of Lyapunov, which shows the closed-loop estimation/tracking ultimate bound behavior. Additionally, numerical simulations and experimental tests validate the robustness of the approach in trajectory-tracking tasks. Full article

The description of the Earth’s gravitational field, governed by the fundamental potential equation (the Laplace equation), is conventionally expressed using spherical harmonics, yet the Cartesian formulation, using a Taylor series representation, offers significant algebraic advantages. This paper proposes a novel Direct Cartesian Method for generating spherical basis functions and coefficients directly within the Cartesian coordinate system, utilising the partial derivatives of the inverse distance ($1/R$) function. The present study investigates the structural correspondence between the Cartesian form of spherical basis functions and the high-order partial derivatives of $1/R$. The study reveals that spherical basis functions can be categorised into four distinct groups based on the parity of the degree n and order m. It is demonstrated that each spherical basis function is equivalent to a weighted summation of the partial derivatives of the inverse distance ($1/R$) with respect to Cartesian coordinates. Specifically, the basis functions are combined with those derivatives that share the same order of Z-differentiation and possess matching parities in their orders of differentiation with respect to X and Y. In order to facilitate the practical calculation of these high-degree derivatives, a recursive numerical algorithm has been developed. The method generates the polynomial coefficients for the numerator of the $1/R$ derivatives. A pivotal innovation is the implementation of a step-wise normalization scheme within the recursive relations. The integration of the recursive ratios of global normalization factors (including full Schmidt normalization) into each step of the algorithm effectively neutralises factorial growth, rendering the process immune to numerical overflow. The validity and numerical stability of the proposed method are demonstrated through a detailed step-by-step derivation of a sectorial basis function ($n=8,m=2$). Full article

Climate-induced flooding is a major issue throughout the globe, resulting in damage to infrastructure, loss of life, and the economy. Therefore, there is an urgent need for sustainable flood risk management. This paper assesses the effectiveness of the hybrid defense system using advanced artificial intelligence (AI) techniques. A data series of energy dissipation (ΔE), flow conditions, roughness, and vegetation density was collected from literature and laboratory experiments. Out of the selected 136 data points, 80 points were collected from literature and 56 from a laboratory experiment. Advanced AI models like Random Forest (RF), Extreme Boosting Gradient (XGBoost) with Particle Swarm Optimization (PSO), Support Vector Regression (SVR) with PSO, and artificial neural network (ANN) with PSO were trained on the collected data series for predicting floodwater energy dissipation. The predictive capability of each model was evaluated through performance indicators, including the coefficient of determination (R²) and root mean square error (RMSE). Further, the relationship between input and output parameters was evaluated using a correlation heatmap, scatter pair plot, and HEC-contour maps. The results demonstrated the superior performance of the Random Forest (RF) model, with a high coefficient of determination (R² = 0.96) and a low RMSE of 3.03 during training. This superiority was further supported by statistical analyses, where ANOVA and t-tests confirmed the significant performance differences among the models, and Taylor’s diagram showed closer agreement between RF predictions and observed energy dissipation. Further, scatter pair plot and HEC-contour maps also supported the result of SHAP analysis, demonstrating greater impact of the roughness condition followed by vegetation density in reducing floodwater energy dissipation under diverse flow conditions. The findings of this study concluded that RF has the capability of modeling flood risk management, indicating the role of AI models in combination with a hybrid defense system for enhanced flood risk management. Full article