Computer Algebra, Power Series and Summation

2020 ◽  
pp. 113-136
Author(s):  
Wolfram Koepf
Keyword(s):  
Power Series ◽  
Computer Algebra

scholarly journals Trinomial equation: the Hypergeometric way

2021 ◽  
Vol 5 (1) ◽  
pp. 236-247
Author(s):  
Daniele Ritelli ◽  
◽  
Giulia Spaletta ◽  
Keyword(s):  
Nineteenth Century ◽  
Power Series ◽  
Computer Algebra ◽  
Free Parameter ◽  
Elliptic Functions ◽  
Late Nineteenth Century ◽  
Algebraic Equations ◽  
Analytical Treatment ◽  
Late Nineteenth ◽  
Trinomial Equation

This paper is devoted to the analytical treatment of trinomial equations of the form \(y^n+y=x,\) where \(y\) is the unknown and \(x\in\mathbb{C}\) is a free parameter. It is well-known that, for degree \(n\geq 5,\) algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [<a href="#1">1</a>,<a href="#2">2</a>] for example) requires the use of power series, following the seminal work of Lagrange [<a href="#3">3</a>]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance.

scholarly journals Power series in computer algebra

1992 ◽  
Vol 13 (6) ◽  
pp. 581-603 ◽  
Author(s):  
Wolfram Koepf
Keyword(s):  
Power Series ◽  
Computer Algebra

scholarly journals COMPUTING THE STABILITY BOUNDARIES FOR THE LAGRANGE TRIANGULAR SOLUTIONS IN THE ELLIPTIC RESTRICTED THREE‐BODY PROBLEM

2006 ◽  
Vol 11 (1) ◽  
pp. 95-104 ◽  
Author(s):  
A. N. Prokopenya
Keyword(s):  
Power Series ◽  
Small Parameter ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Body Problem ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Stability Boundaries ◽  
The Stability ◽  
Three Body

An algorithm is proposed for analytical computing the stability boundaries of the Lagrange triangular solutions in the elliptic restricted three‐body problem. It is based on the infinite determinant method. The algorithm has been implemented by using the computer algebra system Mathematica and the stability boundaries have been determined in the form of power series with respect to a small parameter with accuracy up to the 10th order.

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.

The Software «MMI–calibration 3.0»

Metrologiya ◽  
2020 ◽  
pp. 16-24
Author(s):  
Alexandr D. Chikmarev
Keyword(s):  
Power Series ◽  
Data Processing ◽  
Statistical Treatment ◽  
Correction Function ◽  
Calibration Function ◽  
Measuring Instrument ◽  
Working Standard ◽  
Calibration Protocol ◽  
Error Probabilities ◽  
Continuous Power

A single program has been developed to ensure that the final result of the data processing of the measurement calibration protocol is obtained under normal conditions. The calibration result contains a calibration function or a correction function in the form of a continuous sedate series and a calibration chart based on typical additive error probabilities. Solved the problem of the statistical treatment of the calibration protocol measuring in normal conditions within a single program “MMI–calibration 3.0” that includes identification of the calibration function in a continuous power series of indications of a measuring instrument and chart calibration. An example of solving the problem of calibration of the thermometer by the working standard of the 3rd grade with the help of the “MMI-calibration 3.0” program.

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