scholarly journals On Jacob's construction of the rational canonical form of a matrix

2020 ◽  
Vol 36 (36) ◽  
pp. 177-182
Author(s):  
Meinolf Geck
Keyword(s):  
Normal Form ◽  
Canonical Form ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Efficient Implementation ◽  
Linear Map ◽  
Pure Matrix

H.G. Jacob's elegant approach to the rational canonical, or Frobenius normal form of a linear map is presented here in pure matrix language, thereby avoiding the abstract machinery and prerequisites in the original paper. Related algorithmic aspects and an efficient implementation in the computer algebra system GAP are also discussed.

SIMPLEST NORMAL FORMS OF HOPF AND GENERALIZED HOPF BIFURCATIONS

1999 ◽  
Vol 09 (10) ◽  
pp. 1917-1939 ◽  
Author(s):  
P. YU
Keyword(s):  
Normal Form ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Normal Forms ◽  
Amplitude Equation ◽  
Polar Coordinates ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Odd Order ◽  
Form Theory

The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches. It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal form. In this paper, three theorems are presented to show that the conventional normal forms of Hopf and generalized Hopf bifurcations can be further simplified. The forms obtained in this paper for Hopf and generalized Hopf bifurcations are shown indeed to be the "simplest", and at most only two terms remain in the amplitude equation of the "simplest normal form" up to any order. An example is given to illustrate the applicability of the theory. A computer algebra system using Maple is used to derive all the formulas and verify the results presented in this paper.

Recognition of unimodal map germs from the plane to the plane by invariants

2018 ◽  
Vol 28 (07) ◽  
pp. 1199-1208
Author(s):  
Saima Aslam ◽  
Muhammad Ahsan Binyamin ◽  
Gerhard Pfister
Keyword(s):  
Normal Form ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Closed Field ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Algebraically Closed Field

In this paper, we characterize the classification of unimodal maps from the plane to the plane with respect to [Formula: see text]-equivalence given by Rieger in terms of invariants. We recall the classification over an algebraically closed field of characteristic [Formula: see text]. On the basis of this characterization, we present an algorithm to compute the type of the unimodal maps from the plane to the plane without computing the normal form and also give its implementation in the computer algebra system Singular.

scholarly journals An Implementation of Lipschitz Simple Functions in Computer Algebra System Singular

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Yanan Liu ◽  
Muhammad Ahsan Binyamin ◽  
Adnan Aslam ◽  
Minahal Arshad ◽  
Chengmei Fan ◽  
...  
Keyword(s):  
Normal Form ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Complete Classification ◽  
Simple Function ◽  
Complex Numbers ◽  
Lipschitz Equivalence

A complete classification of simple function germs with respect to Lipschitz equivalence over the field of complex numbers ℂ was given by Nguyen et al. The aim of this article is to implement a classifier in terms of easy computable invariants to compute the type of the Lipschitz simple function germs without computing the normal form in the computer algebra system Singular.

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 Maple computer algebra system: A review

1995 ◽  
Vol 10 (3) ◽  
pp. 329-337 ◽  
Author(s):  
John Hutton ◽  
James Hutton
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
System A
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