Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points

In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square − 1,1 2 . And, we prove that the growth order of the Lebesgue constant is O n + 2 2 . This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is O n . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square − 1,1 2 , the growth order of which is O ln n 2 .

An extension of Lagrange interpolation to approximate derivative, integral and bivariate function

Lagrange interpolation is the effective method to approximate an arbitrary function by a polynomial. But there is a need to estimate derivative and integral given a set of points. Although it is possible to make Lagrange interpolation first, which produces Lagrange polynomial; after that we take derivative or integral on such polynomial. However this approach does not result out the best estimation of derivative and integral. This research proposes a different approach that makes approximation of derivative and integral based on point data first, which in turn applies Lagrange interpolation into the approximation. Moreover, the research also proposes an extension of Lagrange interpolation to bivariate function, in which interpolation polynomial is converted as two-variable polynomial.

Mathematical Analysis of the TB Model with Treatment via Caputo-Type Fractional Derivative

In this study, we formulate a noninteger-order mathematical model via the Caputo operator for the transmission dynamics of the bacterial disease tuberculosis (TB) in Khyber Pakhtunkhwa (KP), Pakistan. The number of confirmed cases from 2002 to 2017 is considered as incidence data for the estimation of parameters or to parameterize the model and analysis. The positivity and boundedness of the model solution are derived. For the dynamics of the tuberculosis model, we find the equilibrium points and the basic reproduction number. The proposed model is locally and globally stable at disease-free equilibrium, if the reproduction number ℛ 0 < 1 . Furthermore, to examine the behavior of the various parameters and different values of fractional-order derivative graphically, the most common iterative scheme based on fundamental theorem and Lagrange interpolation polynomial is implemented. From the numerical result, it is observed that the contact rate and treatment rate have a great impact on curtailing the tuberculosis disease. Furthermore, proper treatment is a key factor in reducing the TB transmission and prevalence. Also, the results are more precise for lower fractional order. The results from various numerical plots show that the fractional model gives more insights into the disease dynamics and on how to curtail the disease spread.

On Ordinary Words of Standard Reed–Solomon Codes over Finite Fields

Reed–Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm. Usually we use the maximum likelihood decoding (MLD) algorithm in the decoding process of Reed–Solomon codes. MLD algorithm relies on determining the error distance of received word. Dür, Guruswami, Wan, Li, Hong, Wu, Yue and Zhu et al. got some results on the error distance. For the Reed–Solomon code [Formula: see text], the received word [Formula: see text] is called an ordinary word of [Formula: see text] if the error distance [Formula: see text] with [Formula: see text] being the Lagrange interpolation polynomial of [Formula: see text]. We introduce a new method of studying the ordinary words. In fact, we make use of the result obtained by Y.C. Xu and S.F. Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed–Solomon codes over the finite field of [Formula: see text] elements. This completely answers an open problem raised by Li and Wan in [On the subset sum problem over finite fields, Finite Fields Appl. 14 (2008) 911–929].

Some Mathematical Aspects of Constructing Artificial Neural Networks

Искусственные нейронные сети (ИНС) в настоящее время являются полем интенсивных исследований. Они зарекомендовали себя при решении задач распознавания образов, аудио и текстовой информации. Планируется их применение в медицине, в беспилотных автомобилях и летательных аппаратах. Однако крайне мало научных работ посвящено обсуждению возможности построения искусственного интеллекта (ИИ), способного эффективно решать очерченный круг задач. Отсутствует гарантия штатного функционирования ИИ в любой реальной, а не специально созданной ситуации. В данной работе предпринимается попытка обоснования ненадежности функционирования современных искусственных нейронных сетей. Показывается, что задача построения интерполяционных многочленов является прообразом проблем, возникающих при создании ИНС. Известны примеры К.Д.Т. Рунге, С.Н. Бернштейна и общая теорема Фабера о том, что для любого наперед заданного натурального числа, соответствующего количеству узлов в интерполяционной таблице, найдется точка из области интерполяции и непрерывная функция, что интерполяционный многочлен не сходится к значению функции в этой точке при неограниченном росте числа узлов. Отсюда следует невозможность обеспечения эффективной работы ИИ лишь за счет неограниченного роста числа нейронов и объемов данных (Big Data), используемых в качестве обучающих выборок. Artificial neural networks (ANN) are currently a field of intensive research. They are a proven pattern/audio/text recognition tool. ANNs will be used in medicine, autonomous vehicles, and drones. Still, very few works discuss building artificial intelligence (AI) that can effectively solve the mentioned problems. There is no guarantee that AI will operate properly in any reallife, not simulated situation. In this work, an attempt is made to prove the unreliability of modern artificial neural networks. It is shown that constructing interpolation polynomials is a prototype of the problems associated with the ANN generation. There are examples by C.D.T. Runge, S.N. Bernstein, and the general Faber theorem stating that for any predetermined natural number corresponding to the number of nodes in the lookup table there is a point from the interpolation region and a continuous function that the interpolation polynomial does not converge to the value of the function at this point as the number of nodes increases indefinitely. This means the impossibility of ensuring efficient AI operation only by an unlimited increase in the number of neurons and data volumes (Big Data) used as training datasets.

Numerical Approximation of One- and Two-Dimensional Coupled Nonlinear Schrödinger Equation by Implementing Barycentric Lagrange Interpolation Polynomial DQM

In this paper, a new numerical method named Barycentric Lagrange interpolation-based differential quadrature method is implemented to get numerical solution of 1D and 2D coupled nonlinear Schrödinger equations. In the present study, spatial discretization is done with the aid of Barycentric Lagrange interpolation basis function. After that, a reduced system of ordinary differential equations is solved using strong stability, preserving the Runge-Kutta 43 method. In order to check the accuracy of the proposed scheme, we have used the formula of L ∞ error norm. The matrix stability analysis method is implemented to test the proposed method’s stability, which confirms that the proposed scheme is unconditionally stable. The present scheme produces better results, and it is easy to implement to obtain numerical solutions of a class of partial differential equations.

More 1-cocycles for classical knots

Let [Formula: see text] be the topological moduli space of long knots up to regular isotopy, and for any natural number [Formula: see text] let [Formula: see text] be the moduli space of all [Formula: see text]-cables of framed long knots which are twisted by a string link to a knot in the solid torus [Formula: see text]. We upgrade the Vassiliev invariant [Formula: see text] of a knot to an integer valued combinatorial 1-cocycle for [Formula: see text] by a very simple formula. This 1-cocycle depends on a natural number [Formula: see text] with [Formula: see text] as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on [Formula: see text] already for [Formula: see text].

An Investigation on using Lagrange, Newton and Least Square Methods for Generating Nonlinear Interpolation Function for the Measuring Instruments

This research is considered the milestone for metrologists to choose the appropriate method for determination of the nonlinear interpolation function for the measuring instruments. Three methods of generating the interpolation polynomial equations were investigated; Newton, Lagrange, and Least Square method. The response of the measuring instruments under investigation was calculated and compared with the experimental results. Least Square method was found that it is the most accurate and most realistic approach to determine the interpolation polynomial function for the measuring instruments. It is recommended to use Least Square method rather than other methods to interpolating the polynomial equation. This recommendation is very important for metrologist as well as for measuring instruments applicant. This article is millstone to determine the response of the measuring instrument at non calibrated points in the calibrated range.

Technology for restoring functional dependencies to determine reliability parameters

The problem of determining the properties of the object by analyzing the numerical and qualitative characteristics of a discrete sample is considered. A method has been developed to determine the probability of trouble-free operation of electronic systems for the case if the interpolation fields are different between several interpolation nodes. A method has been developed to determine the probability of trouble-free operation if the interpolation polynomial is the same for the entire interpolation domain. It is shown that local interpolation methods give more accurate results, in contrast to global interpolation methods. It is shown that in the case of global interpolation it is possible to determine the value of the function outside the given values by extrapolation methods, which makes it possible to predict the probability of failure. It is shown that the use of approximation methods to determine the probability of trouble-free operation reduces the error of the second kind. A method for analyzing the qualitative characteristics of functional dependences has been developed, which allows us to choose the optimal interpolation polynomial. With sufficient statistics, using the criteria of consent, it is possible to build mathematical models for the analysis of failure statistics of electronic equipment. Provided that the volume of statistics is not large, such statistics may not be sufficient and the application of consent criteria will lead to unsatisfactory results. Another approach is to use an approximation method that is applied to statistical material that was collected during testing or controlled operation. In this regard, it is extremely important to develop a method for determining the reliability of electronic systems in case of insufficiency of the collected statistics of failures of electronic equipment.