Taylor Series Research Articles

Optimized Economic Production Quantity Inventory Model with Weibull Decay, Dynamic Holding Cost Under Varied Demand Scenario and Demand Series Analysis

This paper presents a sophisticated modification of the economic production quantity (EPQ) model. It incorporates Weibull-distributed progressive deterioration, dynamic holding costs, variable demand, and strong strategies for effectively handling shortages and backlogs. Targeting products that have a short-term expiration but can still be stored for a long time without significant deterioration, the model aims to increase production quantities while lowering overall variable expenses. The extended EPQ model utilizes Maclaurin series approximations to solve ordinary differential equations, effectively capturing the dynamics of inventory under three different demand patterns. The focus of this handle’s situations when the demand for a product changes in a predictable manner over time and considers the likelihood of defective goods during the manufacturing process. This approach provides useful insights for optimizing inventory management and production planning in the manufacturing of goods that are used up or consumed. The usefulness of the model is confirmed in the study through the use of two illustrative instances. Additionally, sensitivity analysis is utilized to suggest potential areas for additional investigation.

Self-propulsion performance prediction in calm water based on RANS/TEBEM coupling method

This paper proposes a hybrid RANS/TEBEM method to conduct self-propulsion prediction in calm water with accuracy and improve its efficiency by means of larger time step. In the coupling procedure, the rotating propeller is represented by time-averaged, spatial inhomogeneous body force field obtained from the potential flow solver based on Taylor Expansion Boundary Element Method (TEBEM) in RANS simulation. Effective wake is evaluated by subtracting induced velocities computed by potential solver from total velocities in RANS solver at a coupling plane upstream of propeller plane. The influence of coupling plane position and grid density on the numerical results is also investigated. Through KCS and KVLCC2 cases, it is shown that the predicted value of propeller revolution speed is within 4% of the experimental value, verifying the robustness and reliability of this method.

Majoration of the modulus of the Schwarzian of certain quasi-conformal maps

In this work, we have proposed a majoration of the modulus of the Schwarzian of certain quasi-conformal maps, defined in the open unit disk ∆ = z ∈ C : |z| < 1}. Using the Schwarzian, we have found some relations linking the Taylor coefficients of these maps.

Strong Energy Conversion by a Ribbon‐Like Magnetic Structure in Tailward High‐Speed Flow

AbstractThe tailward high‐speed flows, in which various kinetic processes and magnetic structures can be embedded, are usually produced by the magnetic reconnection in the Earth's magnetotail. Here, using high‐resolution measurements from the Magnetospheric Multiscale (MMS) mission, we report an intense current in the tailward high‐speed flow. The current density can reach ∼180 nA/m2 and is primarily carried by electrons. Taking advantage of the First‐Order Taylor Expansion (FOTE) method, we reveal that such an intense current is associated with a ribbon‐like magnetic structure. A large electric field, reaching ∼90 mV/m, is also observed at the magnetic structure. The Hall term dominates the electric field, however, the contribution from the pressure gradient term and the electron inertial term is nonnegligible and can lead to strong energy conversion (E ⋅ J > 2 nW/m3) through the synergistic action with the intense current. This study improves the understanding of the current behaviors and energy conversion associated with magnetic structures in the Earth's magnetotail.

Topological defects on mass spectra of quarkonium system with a Class of Yukawa potential (CYP)

In this paper, our focus lies in investigating nonrelativistic quantum systems under external potentials by solving the Schrödinger wave equation within a topological defect background. We opt for a class of Yukawa potential (CYP) and substitute the screening parameter with the Debye mass, which varies with temperature. Employing a Taylor series expansion to simplify the exponential term, we solve the radial wave equation, yielding an analytical solution for the quantum system in question, referred to as the Quarkonium system. Our analysis reveals that the presence of a topological defect modifies the energy profile of the Quarkonium system, consequently disrupting the degeneracy of the energy spectra.

Frequent Social Media Use and Experiences with Bullying Victimization, Persistent Feelings of Sadness or Hopelessness, and Suicide Risk Among High School Students - Youth Risk Behavior Survey, United States, 2023.

Social media has become a pervasive presence in everyday life, including among youths. In 2023, for the first time, CDC's nationally representative Youth Risk Behavior Survey included an item assessing U.S. high school students' frequency of social media use. Data from this survey were used to estimate the prevalence of frequent social media use (i.e., used social media at least several times a day) among high school students and associations between frequent social media use and experiences with bullying victimization, persistent feelings of sadness or hopelessness, and suicide risk. All prevalence estimates and measures of association used Taylor series linearization. Prevalence ratios were calculated using logistic regression with predicted marginals. Overall, 77.0% of students reported frequent social media use, with observed differences by sex, sexual identity, and racial and ethnic identity. Frequent social media use was associated with a higher prevalence of bullying victimization at school and electronically, persistent feelings of sadness or hopelessness, and some suicide risk among students (considering attempting suicide and having made a suicide plan), both overall and in stratified models. This analysis characterizes the potential harms of frequent social media use for adolescent health among a nationally representative sample of U.S. high school students. Findings might support multisectoral efforts to create safer digital environments for youths, including decision-making about social media policies, practices, and protections.

A new perspective on optimizers: leveraging moreau-yosida approximation in gradient-based learning

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

About One Multistep Multiderivative Method of Predictor-Corrector Type Constructed for Solving İnitial-Value Problem for ODE of Second Order

Considering the wide application of the initial-value problem for Ordinary Differential Equations second-order with a special structure, here for solving this problem constructed the special Multistep Multiderivative Methods. Many scientists studied this problem , but the most distinguishing is the Ştörmer. To solve this problem here is proposed to use the Multistep Second derivative Method with a special structure. This method has been generalized by many authors, which is called as the linear Multistep Multiderivative Methods with the constant coefficients. Many authors shave shown that the Multistep Second derivative Method can be applied to solve the initial-value problem for ODEs of the first order. Euler himself using his famous method discovered that, in his method when moving from one point to another local truncation errors add up, the results of which reach a very large value. To solve this problem, he suggested using more accurate methods. For this aim, Euler proposed calculating the next term in the Taylor series of the solutions of the investigated problem. Developing this idea and papulation of the Multistep Multiderivative Methods here to solve the named problem it is suggested to use MultistepThriedderivative Methods, taking into account that methods of this type are more accurate. For the demonstration above, receiving results here have constructed some concrete methods. Also by using some of Dahlquist’s and Ibrahimov’s results for Multistep Methods with the maximum order of accuracy were compared. Proven that the MultistepThriedderivative Methods are more accurate than the others. By using model problems have illustrated some results received here.

Research on the prediction method of wing structure noise based on the combination of conditional generative adversarial neural network and numerical methods

This paper presents a technique for predicting noise generated by airfoil structures that combines deep learning techniques with traditional numerical methods. In traditional numerical methods, accurately predicting the noise of airfoil structures requires significant computational resources, making it challenging to perform low-noise optimization design for these structures. To expedite the prediction process, this study introduces Conditional Generative Adversarial Networks (CGAN). By replacing the generator and discriminator of CGAN with traditional regression neural network models, the suitability of CGAN for regression prediction is ensured. In this study, the data computation was accelerated by expanding the kernel function in the traditional boundary element method using a Taylor series. Based on the resulting data, an alternative predictive model for wing structure noise was developed by integrating Conditional Generative Adversarial Networks (CGAN). Finally, the effectiveness and feasibility of the proposed method are demonstrated through three case studies.

NONMONOGENITY OF NUMBER FIELDS DEFINED BY TRUNCATED EXPONENTIAL POLYNOMIALS

Abstract Let p be a prime number. Let $n\geq 2$ be an integer given by $n = p^{m_1} + p^{m_2} + \cdots + p^{m_r}$ , where $0\leq m_1 < m_2 < \cdots < m_r$ are integers. Let $a_0, a_1, \ldots , a_{n-1}$ be integers not divisible by p. Let $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta \in {\mathbb C}$ a root of an irreducible polynomial $f(x) = \sum _{i=0}^{n-1}a_i{x^i}/{i!} + {x^n}/{n!}$ over the field $\mathbb Q$ of rationals. We prove that p divides the common index divisor of K if and only if $r>p$ . In particular, if $r>p$ , then K is always nonmonogenic. As an application, we show that if $n \geq 3$ is an odd integer such that $n-1\neq 2^s$ for $s\in {\mathbb Z}$ and K is a number field generated by a root of a truncated exponential Taylor polynomial of degree n, then K is always nonmonogenic.

Comparative analysis of macroscopic traffic models for evaluating shockwaves under lane closures on an urban highway using the LWR model

ABSTRACT Lane closure due to traffic crashes is a complex phenomenon that negatively affects the operational performance of highways and leads to secondary lane closures. The present study investigates the effects of shockwaves caused by a heavy vehicle run-off-road crash on a three-lane urban highway in Isfahan, Iran. Additionally, the study simulates the effects of lane closures on shockwaves and density using the Lighthill, Whitham, and Richards (LWR) theory and macroscopic traffic flow based on field data. An analysis of traffic flow models revealed that one-lane and two-lane closures reduce maximum flow by 29% and 61%, respectively, compared to normal conditions. The results showed that the minimum relative error (RE) for lane closures is associated with shockwave models based on the Underwood speed-density model, expanded using the Taylor series. Therefore, the shockwave model based on the Underwood model with Taylor series expansion outperforms other models.

Pruning convolutional neural networks for inductive conformal prediction

Neural network pruning is a popular approach to reducing model storage size and inference time by removing redundant parameters in the neural network. However, the uncertainty of predictions from pruned models is unexplored. In this paper we study neural network pruning in the context of conformal predictors (CP). The conformal prediction framework built on top of machine learning algorithms supplements their predictions with reliable uncertainty measure in the form of prediction sets, under the independent and identically distributed assumption on the data. Convolutional neural networks (CNNs) have complicated architectures and are widely used in various applications nowadays. Therefore, we focus on pruning CNNs and, in particular, filter-level pruning. We first propose a brute force method that estimates the contribution of a filter to the CP’s predictive efficiency and removes those with the least contribution. Given the computation inefficiency of the brute force method, we also propose the Taylor expansion to approximate the filter’s contribution. Furthermore, we improve the global pruning method by protecting the most important filters within each layer from being pruned. In addition, we explore the ConfTr loss function which is optimized to yield maximal CP efficiency in the context of neural network pruning. We have conducted extensive experimental studies and compared the results regarding the trade-offs between predictive efficiency, computational efficiency, and network sparsity. These results are instructive for deploying pruned neural networks with applications using conformal prediction where reliable predictions and reduced computational cost are relevant, such as in safety-critical applications.

Simplified inverse distance weighting-immersed boundary method for simulation of fluid-structure interaction

When simulating fluid-structure interaction (FSI) problems involving moving objects, the implicit inverse distance weighting-immersed boundary method (IDW-IBM) developed by Du et al. [1] has to construct a large square correlation matrix and solve its inversion at each time step. In this work, a simplified inverse distance weighting-immersed boundary method (SIDW-IBM) is proposed to eliminate the intrinsic limitations in the original implicit IDW-IBM. Through error analysis using Taylor series expansion, a second order approximation can be derived, which allows us to approximate the large square correlation matrix into a diagonal matrix; thereby, we proposed the SIDW-IBM based on this second order approximation to explicitly evaluate the velocity corrections, where the needs to assemble the large correlation matrix and inverse it are circumvented. Owing to the fact that the inverse distance weighting interpolation removes the limitations in the Dirac delta function, the proposed SIDW-IBM has been successfully implemented on the non-uniform meshes to further improve the computational efficiency. The proposed SIDW-IBM is integrated with the reconstructed lattice Boltzmann flux solver (RLBFS) [2] to simulate some classic incompressible viscous flows, including flow past an in-line oscillating cylinder, flow past a heaving airfoil, and flow past a three-dimensional flexible plate. The good agreement between the present results and reference data demonstrates the capability and feasibility of the SIDW-IBM for simulating FSI problems with moving boundaries and large deformations.

A reduced-order hybrid model for photobioreactor performance and biomass prediction

This paper introduces a hybrid approach for photobioreactor modeling tailored to microalgae cultivation, combining data-driven and mechanistic concepts to improve modeling efficiency and practicality for industrial scale-up applications. Most growth models for microalgae are nonlinear and require experimental measurement of several parameters. The aim of this work is to develop linear practical models for monitoring purposes. A model based on linear coefficients and polynomial features is proposed, balancing interpretability with non-linear representation focusing on model transparency. To simplify the growth model, Taylor series expansion is applied to the Monod and logistic population models. Two scale-specific models are developed and evaluated, offering practical solutions for monitoring microalgae growth in photobioreactors. Therefore, this reduced order representation allows the biomass growth rate to be dependent directly on the biomass concentration. These models do not require exhaustive data collection of substrate concentration over time, making them cost-effective and efficient for industrial applications. This work provides a step forward in photobioreactor modeling, contributing to the sustainable production of microalgae.

A Shifted Boundary Method for the compressible Euler equations

The Shifted Boundary Method (SBM) is applied to compressible Euler flows, with and without shock discontinuities. The SBM belongs to the class of unfitted (or immersed, or embedded) finite element methods and avoids integration over cut cells (and the associated implementation/stability issues) by reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. We specifically discuss the advantages the proposed method offers in avoiding spurious numerical artifacts in two scenarios: (a) when curved boundaries are represented by body-fitted polygonal approximations and (b) when the Kutta condition needs to be imposed in immersed simulations of airfoils. An extensive suite of numerical tests is included.

Enhanced high-frequency continualization scheme for inertial beam-lattice metamaterials

A multifield continualization technique is introduced that offers a thermodynamically consistent description of the constitutive and dispersive properties of beam-lattice inertial metamaterials with periodic microstructures. The balance equations governing the mechanics of the discrete Lagrangian system are appropriately handled using an innovative continualization scheme to derive an equivalent integral-type continuum model. Based on formal Taylor series expansion of the integral kernels or the corresponding pseudo-differential functions incorporating shift operators and appropriate pseudo-differential downscaling laws, the proposed multifield enhanced continualization scheme allows the derivation of a gradient-type continuum model of given rank and equivalent to lattices. Two different resolution techniques are proposed. Firstly, the corresponding infinite-order average differential equations are tackled using a perturbative approach to describe the forced Bloch wave propagation in the metamaterial. Secondly, higher-order continuum models are employed through proper differential equation truncation to characterize the dispersive properties of the metamaterial in both high- and low-frequency regimes. Moreover, an energetically consistent generalized equivalent Micropolar continua, with non-local inertial terms, are here identified. The multifield continualization procedure is applied to two-dimensional periodic microstructures with tetrachiral, hexachiral, and hexa-tetrachiral topologies. Illustrative examples highlight the ability of the equivalent continuum model to accurately describe the effective constitutive properties of inertial metamaterials with periodic microstructures and to define a dynamic response consistent with the discrete Lagrangian model, validated and tested through virtual experimental verification under free and forced wave conditions.

Some Coefficient Problems for Subclasses of Holomorphic Functions in Complex Order Associating Sălăgeăn q-Differential Operator

A function with complex values and at every point of the specific domain contains a derivative is commonly known as analytic functions which is also referred as holomorphic functions. We begin by interpreting \(A\) as the class for all holomorphic functions \(L(\xi)\) with a Taylor series expansion written in the form: \[L(\xi) = \xi + \sum_{i=2}^{\infty} x_i \xi^i\] where \(x_i \in \mathbb{C}\) and \(\xi \in D\). \(D\) is the open unit disk where \(D = \{\xi : \xi \in \mathbb{C}, |\xi| < 1 \}\). Furthermore, we suggest the subclass of \(A\) that is univalent in \(D\) represented as \(S\). It is commonly known that the main subclasses of class \(S\) are the class of starlike functions and the class of convex functions. To develop and analyze the Fekete-Szegö problems, the theory of geometric function contributes significantly to this. Moreover, the frequent use of \(q\)-calculus as a general direction of research among mathematicians has caught our attention. In this research, we attained the initial coefficients, \(x_2\) and \(x_3\), and the upper bound for the functional \(|x_3 - \nu x_2^2|\) of functions \(L\) in the two new subclasses that are introduced by involving the Sălăgeăn \(q\)-differential operator, \(M_q^\eta L(\xi)\) and the definition of subordination.

Accuracy and Computational Cost Comparison of Numerical Flight Dynamics Reduced-Order Models

This paper compares several approaches to model the unsteady and nonlinear longitudinal aerodynamics of a generic unmanned combat air vehicle configuration based on computational fluid dynamics (CFD). Three different reduced-order models (ROMs) are implemented through unsteady simulations to predict the evolution of forces and moments during prescribed trajectories carried out at M=0.15. The first model investigates a linear quasi-steady model based on a first-order Taylor series expansion of the aerodynamic coefficients. The second one picks up the mathematical formulation of the first model but adds a dynamic coefficient in order to take into account the unsteady effect of rotation rate variations. Finally, the third model is built on indicial functions computed from positive step changes in the angle of attack or pitch rate. The unknown parameters of the first two models and the unknown indicial functions of the third one are obtained, respectively, from the study of forced oscillations and step motions. These prescribed trajectories are performed using unsteady Reynolds-averaged Navier–Stokes simulations coupled with a grid displacement tool. Two different methods are tested and compared to carry out the CFD mesh motions: an overset method and a full moving grid (FMG) method. The indicial functions obtained using the FMG method give results equivalent to those found in the literature in half the time of the overset method. To reduce the computational costs to set up the ROMs, the FMG method is preferred to the overset method. ROMs are then applied, and their predictions are compared with CFD simulations for prescribed trajectories with different angular rates and angles of attack. A study of instantaneous and overall accuracy associated with the implementation cost of the ROMs has allowed to show their strengths and their weaknesses. Similar results are observed at low angles of attack for the ROMs studied, while a major modeling advantage of the indicial method has been identified for higher angles of attack in the nonlinear domain.

Far-field radiation patterns of distributed acoustic sensing in anisotropic media with an explosive source and vertically straight fiber

Distributed acoustic sensing (DAS) is increasingly used in seismic exploration owing to its wide frequency range, dense sampling and real-time monitoring. DAS radiation patterns help to understand angle response of DAS records and improve the quality of inversion and imaging. In this paper, we solve the 3D vertical transverse isotropic (VTI) Christoffel equation and obtain the analytical, first-order, and zero-order Taylor expansion solutions that represent P-, SV-, and SH-wave phase velocities and polarization vectors. These analytical and approximated solutions are used to build the P/S plane-wave expression identical to the far-field term of seismic wave, from which the strain rate expressions are derived and DAS radiation patterns are thus extracted for anisotropic P/S waves. We observe that the gauge length and phase angle terms control the radiating intensity of DAS records. Additionally, the Bond transformation is adopted to derive the DAS radiation patterns in title transverse isotropic (TTI) media, which exhibits higher complexity than that of VTI media. Several synthetic examples demonstrate the feasibility and effectiveness of our theory.

The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensor

Given a Cauchy surface in a curved spacetime and a suitably defined quantum state on the CCR algebra of the Klein-Gordon quantum field on that surface, we show, by expanding the squared spacetime geodesic distance and the 'U' and 'V' Hadamard coefficients (and suitable derivatives thereof) in sufficiently accurate covariant Taylor expansions on the surface that the renormalized expectation value of the quantum stress-energy tensor on the surface is determined by the geometry of the surface and the first 4 time derivatives of the metric off the surface, in addition to the Cauchy data for the field's two-point function. This result has been anticipated in and is motivated by a previous investigation by the authors on the initial value problem in semiclassical gravity, for which the geometric initial data corresponds, a priori, to the spatial metric on the surface and up to 3 time derivatives off the surface, but where it was argued that the fourth derivative can be obtained with aid of the field equations on the initial surface.