A Generalization of the Taylor Series to Rational Functions and Walsh Equiconvergence

2006 ◽  
pp. 55-80
Keyword(s):  
Taylor Series ◽  
Rational Functions

On Arithmetic Properties of the Taylor Series of Rational Functions

1969 ◽  
Vol 21 ◽  
pp. 378-382 ◽  
Author(s):  
David G. Cantor
Keyword(s):  
Rational Function ◽  
Taylor Series ◽  
Rational Functions ◽  
Complex Numbers ◽  
Finitely Generated ◽  
Algebraic Integers ◽  
Common Generalization ◽  
Arithmetic Properties ◽  
Complex Coefficients

Pólya (3) has shown that if bnis a sequence of algebraic integers and is a rational function, then so is . This result was generalized by Uchiyama (5) who showed that one may replace the assumption that the bn are algebraic integers by the assumption that the bn lie in a finitely generated submodule of the complex numbers, and by the author (1) who showed that if p is a non-zero polynomial with complex coefficients and if bn is a sequence of algebraic integers such that is a rational function, then so is . Our aim in this note is to give a common generalization of all of these theorems.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

Investigation of Accuracy of a Calibrated Estimator of a Ratio by Modelling

2005 ◽  
Vol 10 (4) ◽  
pp. 333-342
Author(s):  
V. Chadyšas ◽  
D. Krapavickaitė
Keyword(s):  
Mean Square Error ◽  
Taylor Series ◽  
Random Sampling ◽  
Finite Population ◽  
Taylor Series Expansion ◽  
Simple Random Sampling ◽  
Population Parameter ◽  
Mean Square ◽  
Parameter Ratio ◽  
Using Data

Estimator of finite population parameter – ratio of totals of two variables – is investigated by modelling in the case of simple random sampling. Traditional estimator of the ratio is compared with the calibrated estimator of the ratio introduced by Plikusas [1]. The Taylor series expansion of the estimators are used for the expressions of approximate biases and approximate variances [2]. Some estimator of bias is introduced in this paper. Using data of artificial population the accuracy of two estimators of the ratio is compared by modelling. Dependence of the estimates of mean square error of the estimators of the ratio on the correlation coefficient of variables which are used in the numerator and denominator, is also shown in the modelling.

Truncation Error for the Simpson’s 1/3 Rule Based on Taylor Series Expansion

2020 ◽  
Author(s):  
Antonio Carlos Foltran ◽  
Carlos Henrique Marchi ◽  
Luís Mauro Moura
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Truncation Error ◽  
Taylor Series Expansion ◽  
Rule Based

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

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