Sum Of Series Research Articles

Characterization of a Large Family of Convergent Series That Leads to a Rapid Acceleration of Slowly Convergent Logarithmic Series

Logarithmic series are known to have a very slow rate of convergence. For example, it takes more than the first 20,000 terms of the sum of the reciprocals of squares of the natural numbers to attain 5 decimal places of accuracy. In this paper, I will devise an acceleration scheme that will yield the same level of accuracy with just the first 400 terms of that power series. To accomplish this, I establish a relationship between all monotonically decreasing sequence of positive terms whose sum converges, a positive number <i>ρ </i>and a differentiable function <i>φ</i>. Then, I use <i>ρ </i>and <i>φ </i>to define the <i>T<sub>φ, ρ </sub></i>transformations on the partial sums of any convergent series. Furthermore, I prove that these <i>T<sub>φ, ρ </sub></i>transformations yield a rapid rate of convergence for many slowly convergent logarithmic series. Finaly, I provide several examples on how to compute <i>φ </i>if one is given the convergent series of decreasing, positive terms.

Approximate solution of a zero-sum linear-quadratic differential game by the auxiliary parameter method: analytical/numerical study

ABSTRACT A zero-sum finite horizon linear-quadratic differential game is considered. First, the terminal-value problem for the game-theoretic Riccati matrix differential equation, associated with the considered game by the solvability conditions, is analysed. This problem may not have the solution in the entire time-interval of the game's duration. Using the method of auxiliary parameter, a sufficient condition (in the terms of the game's data) for the existence of the solution to the aforementioned terminal-value problem in the entire time-interval of the game's duration is derived. An approximate analytical solution to this problem also is obtained as a partial sum of some infinite series of functions arising in the method of auxiliary parameter. Based on this approximate solution, suboptimal state-feedback controls of the players are formally designed and justified. It is shown that the pair of these controls constitutes an approximate-saddle point in the considered game. The theoretical results of the article are illustrated by their application to approximate solution of the problem of pursuit-evasion engagement between two flying vehicles.

2022 개정 교육과정에서 넓이로서의 정적분 정의에 따른 교수학적 난점 및 대안 탐색

This study aims to analyze the expected didactic difficulties and prepare complementary measures for the concept of definite integral, defined as 'the area between the curve and the -axis' in the 2022 Curriculum, following the 2015 Curriculum. To this end, the development of the area concept was examined by comparing the 2022 and 2015 Curricula and analysing textbooks. The analysis revealed difficulties in understanding definite integral as the area due to the limited development of the area concept, such as only justifying the method for calculating the area of a rectangle when the sides are natural numbers. Additionally, it was found that the situation material for questioning that had already been developed in the 2015 Curriculum, introducing the 'relationship between definite integral and sum of series', were ineffective in the 2022 Curriculum. The material were revised and supplemented for use in the 2022 <CalculusII>. Based on this discussion, improvement measures for the area concept development in the 2022 Curriculum were proposed.

Аналіз даних теплового режиму комутаційного обладнання комп’ютерних мереж на основі відновлення сигналів температурних датчиків

The article considers the task of analyzing distributed data of temperature modes of chips of switching equipment of computer networks. For this, a temperature measurement system using a temperature sensor is used. In the case when the temperature sensor is inside the chip, the speed of the measurement system's response to temperature changes is satisfactory. However, when the sensor is outside the chip, due to the inertness of the thermal contact, the response speed of the measuring system is low. In this situation, the effectiveness of the control system becomes unsatisfactory. To overcome the inertness of temperature sensors, it is proposed to analyze the distributed data that is received in digital form on the data processing server by restoring the distorted signals of nonlinear measuring subsystems «chip – temperature sensor» based on the use of their mathematical models in the form of a partial sum of the Volterra integro-power series. The identification of the mathematical model of the measurement subsystem is carried out on a finite interval of time by conducting a series of experiments using test signals. The method of reducing the number of test signals based on taking into account the specificity of the impact of nonlinearity on the results of experiments is considered. The obtained model is the basis for solving the inverse problem of restoring the signal of temperature influence at the sensor input. Since this problem is incorrect, it is suggested to supplement the model with a regularization parameter and reduce the problem to a correct one. To use the model over an infinite period of time, a computer modeling technique is proposed using restarts of computing processes, which are carried out in several streams with a shift in time. The result of calculations is formed by combining fragments from different streams. To check the reliability of the results obtained by applying the developed method, the solutions of the model problems are given.

FDP-FL: differentially private federated learning with flexible privacy budget allocation

Abstract Federated learning (FL) as a privacy-preserving technology enables multiple clients to collaboratively train models on decentralized data. However, transmitting model parameters between local clients and the central server can potentially result in information leakage. Differentially private federated learning (DPFL) has emerged as a promising solution to enhance privacy. Nevertheless, existing DPFL schemes suffer from two issues: (i) most schemes that aim to achieve desired model accuracy may incur a high privacy budget. (ii) several schemes that consider the trade-off between privacy and accuracy by utilizing linear clipping bound may distort numerous model parameters. In this paper, we first propose FDP-FL, a flexible differential privacy approach for FL. FDP-FL introduces a novel series sum privacy budget allocation instead of uniform allocation and enables adaptive and nonlinear noise scale decay. In this way, a tight bound for cumulative privacy loss can be achieved while optimizing model accuracy. Then in order to mitigate gradient leakages caused by honest-but-curious clients and server, we further design client-level FDP-FL and record-level FDP-FL, respectively. Experimental results demonstrate that our FDP-FL improves model accuracy by $\sim $13.3% compared with the basic DP-FL under a fixed privacy budget and outperforms existing trade-off schemes with the same hyperparameter setting.

Sums of singular series along arithmetic progressions and with smooth weights

Sums of the singular series constants that appear in the Hardy–Littlewood [Formula: see text]-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo [Formula: see text] or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo [Formula: see text] of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats.

Calculation of Green's functions for heat conduction in curved anisotropic media

AbstractSolving the inverse problems of heat conduction often requires performing a very large number of Green's function evaluations. This paper addresses the calculation of Green's functions in anisotropic curved media, such as composite lamina, where the principal axes of thermal conductivity follow the curved surface. We start with established exact series solutions for cylindrical convex and concave shapes. These solutions are extended to accommodate geometries of real‐world composite materials that do not form a closed surface like a cylinder. Unfortunately, the exact solutions have the form of an infinite series of sums or integrals and are computationally infeasible for the inverse problems of interest, especially for large radii of curvature. This motivates a perturbation solution that is accurate and computationally efficient at large radii of curvature. In addition, it motivates a phenomenological approximation that is extremely computationally efficient over a broad range of curvatures, but with some sacrifice in accuracy. These solutions are compared with exact solutions, with each other, and with numerical finite difference calculation. We identify regions in parameter space where the different approaches are preferable and where they lead to the same numerical result.

Propagation of the Lower Estimate of the Sum of a Sine Series with Convex Sequence of Coefficients Obtained by Telyakovskii to a Longer Segment

Propagation of the Lower Estimate of the Sum of a Sine Series with Convex Sequence of Coefficients Obtained by Telyakovskii to a Longer Segment

Sums of infinite series involving the Riemann zeta function

Sums of infinite series involving the Riemann zeta function

A method to test the uniform convergence of function series

The series refer to performing infinite addition operations on infinite numbers or functions in a certain order. It is hard to find out whether the positive function series converges uniformly in many cases. In this article, a new method that replacing the sum of function terms series with improper integral will be introduced, which is designed to solve problems that cannot be solved by classical Weierstrass M-test. The Cauchy uniform convergence test will serve as the basis for the entire proof process because it can lead the focus point from the whole sum to the partial sum of the function series, where its value can be easier substituting by the value of the improper integral. After using basic knowledge of the improper integral, the uniform convergence can finally be known. By using this method, testing the uniform convergence of the irregular function series even estimating its value can be possible accomplished.

Enhanced and generalized one–step Neville algorithm: Fractional powers and access to the convergence rate

The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular choice of the abscissas used for the extrapolation, one can replace the recursive Neville scheme by a simple one-step transformation, while also obtaining access to subleading terms for the transformed series after convergence acceleration. The matrix-based, unified formulas allow one to estimate the rate of convergence of the partial sums of the input series to their limit. In particular, Bethe logarithms for hydrogen are calculated to 100 decimal digits. Generalizations of the method to series whose remainder terms can be expanded in terms of inverse factorial series, or series with half-integer powers, are also discussed.

Well-poised [formula omitted]-series and new Fibonomial-like sums

We introduce various sum formulae involving products of two Fibonomial/Lucanomial coefficients or quotients of two such coefficients over a third one. In order to prove the claims we will convert them into their q-forms and then use known terminating well-poised q-series identities.

Seminormed approximation of functions by operators based on their Fourier series

This paper deals with the degree of approximation by an operator based on the partial sum of the Fourier series of a function of class Lp,p≥1. We use modulus of continuity of the function satisfying weaker conditions than those used by Deǧer and Bayindir [On the degree of approximation by the Woronoi-Nörlund and Riesz type means in the GHM, Azerb. J. Math., 8 (2018), 122-141]. The results of our paper also improve those of Singh and Sonker [Degree of approximation of function in Hpw in generalized Hölder metric by matrix means, Mathematical modelling and scientific computation, 1-10, Commun. Comput. Inf. Sci., 283, Springer, Heidelberg, 2012].

Two-dimensional transient contrasting structure evolution in an inhomogeneous media with the advection

The inner transition layer evolution for two-dimensional quasi-linear initial-boundary value problem for the reaction-advection-diffusion equation in an inhomogeneous media with a small parameter within the high order derivatives is considered. Within the framework of the main (zero order) sum of the asymptotic series, the position of the inner transition layer is described by the Hamilton–Jacobi equation. The potential is calculated as an integrated density function of the reaction sources within the limits of the equilibrium levels. The front line of the transition layer evolves in the same way as the constant-eikonal line (the wavefront line in the other words) for the problem of wave propagation in an inhomogeneous medium in short-wave (geometro-optical) asymptotics. The sum of the asymptotic series of zero order and first order is found, the existence gap of a smooth front line, the time of destruction of the contrasting structure are calculated.

Towards a Unified Approach for Continuously-Variable Impedance Control of Powered Prosthetic Legs over Walking Speeds and Inclines.

Research in powered prosthesis control has explored the use of impedance-based control algorithms due to their biomimetic capabilities and intuitive structure. Modern impedance controllers feature parameters that smoothly vary over gait phase and task according to a data-driven model. However, these recent efforts only use continuous impedance control during stance and instead utilize discrete transition logic to switch to kinematic control during swing, necessitating two separate models for the different parts of the stride. In contrast, this paper presents a controller that uses smooth impedance parameter trajectories throughout the gait, unifying the stance and swing periods under a single, continuous model. Furthermore, this paper proposes a basis model to represent intertask relationships in the impedance parameters-a strategy that has previously been shown to improve model accuracy over classic linear interpolation methods. In the proposed controller, a weighted sum of Fourier series is used to model the impedance parameters of each joint as continuous functions of gait cycle progression and task. Fourier series coefficients are determined via convex optimization such that the controller best reproduces the joint torques and kinematics in a reference able-bodied dataset. Experiments with a powered knee-ankle prosthesis show that this simpler, unified model produces competitive results when compared to a more complex hybrid impedance-kinematic model over varying walking speeds and inclines.

Moments of the Nonnegative Adjusted Estimator of Squared Multiple Correlation

I present the moments of the nonnegative adjusted estimator of the squared multiple correlation ρ 2 , the coefficient of determination for random-predictor regression. This estimator, first proposed by Ezekiel, replaces with zero the negative estimates from the well-known adjusted estimator proposed by Fisher that, in turn, corrects the positive bias of the sample R 2. Although Fisher’s version is presented in texts, Ezekiel’s version is used in practice. Each moment comprises a binomial sum of a negative binomial series of incomplete beta functions. Numerical computations, for which an R function is provided, are required to examine the moments. Ezekiel’s estimator is positively biased for smaller ρ 2 and negatively biased for larger. It dominates Fisher’s via MSE. It does not dominate R 2, but the MSE of Ezekiel’s estimator can be substantially smaller but at most negligibly larger. Possible applications to powers of ρ 2 and to other adjusted estimators are briefly discussed.

On calculating partial sums of multiple numerical series by methods of Computer Algebra

A method to calculate partial sums of some multiple numerical series arising when searching for the resultant of a polynomial and an entire function is proposed. One can apply a symbolic algorithm that uses recurrent Newton formulas to find power sums of roots included in this formula without finding the very roots of the system. The algorithm that implements the proposed approach to calculate partial sums of multiple numerical series is implemented in Maple. Examples of using this algorithm to find partial sums of some classes of multiple numerical series are given.

Integrability of series with respect to multiplicative systems and generalized derivatives

We give some necessary and sufficient conditions for the convergence of generalized derivatives of sums of series with respect to multiplicative systems and the corresponding Fourier series. These conditions are counterparts of trigonometric results of S. Sheng, W.O. Bray and C\v .V. Stanojevi´c and extend some results of F. M´oricz proved for Walsh–Fourier series

On Calculating Partial Sums of Multiple Numerical Series by Methods of Computer Algebra

On Calculating Partial Sums of Multiple Numerical Series by Methods of Computer Algebra

Mathematical Modeling of Elastically Deformed States of Thin Isotropic Plates Using Chebyshev Polynomials

Abstract. In this paper a method for solving an inhomogeneous biharmonic equation while modeling elastically deformed states of thin isotropic rectangular plates using a system of orthogonal Chebyshev polynomials of the first kind is proposed. The method is based on representation of a solution to the initial biharmonic equation as a finite sum of Chebyshev series by each independent variable in combination with matrix transformations and properties of Chebyshev polynomials. The problem is examined for the case when a transverse load acts on the plate, and the hinge fastening along the edges of the plate is taken as boundary conditions. Using the extremes and zeros of Chebyshev polynomials of the first kind as collocation points, the boundary value problem is reduced to a system of linear algebraic equations. Decomposition coefficients of desired function with respect to Chebyshev polynomials act as unknowns in this system. As the comparison showed, the results obtained by this method with a high degree of accuracy coincide with similar results derived using analytical approach that are given in the article. The paper also presents the results of calculations using the proposed method in the case when two opposite edges of the plate are pinched and two others are pivotally fixed. The comparison with similar results of modeling the stress-strain states of rectangular plates which are presented in the open sources is carried out.