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.

Primitive Genus Two Groups of Meta Type

2018 ◽  
Vol 7 (1-2) ◽  
pp. 1
Author(s):  
Haval Mohammed Salih
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Complete Classification ◽  
Permutation Groups ◽  
Affine Groups ◽  
Two Systems ◽  
Genus Two

This paper is a contribution to the classification of the finite primitive permutation groups of genus two. We consider the case of affine groups. Our main result, Lemma 3.10 gives a complete classification of genus two systems when . We achieve this classification with the aid of the computer algebra system GAP.

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.

A classifier for simple space curve singularities

2014 ◽  
Vol 51 (1) ◽  
pp. 92-104
Author(s):  
Faira Janjua ◽  
Gerhard Pfister
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Plane Curve ◽  
Space Curve ◽  
Plane Curve Singularities ◽  
Curve Singularities

The classification of Bruce and Gaffney respectively Gibson and Hobbs for simple plane curve singularities respectively simple space curve singularities is characterized in terms of invariants. This is the basis for the implementation of a classifier in the computer algebra system singular.

On the classification of simple maps from the plane to the plane

2017 ◽  
Vol 16 (10) ◽  
pp. 1750199 ◽  
Author(s):  
Muhammad Ahsan Binyamin ◽  
Hasan Mahmood ◽  
Shamsa Kanwal
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System

In this paper, we characterize the classification of simple maps from the plane to the plane given by J. H. Rieger, in terms of invariants. On the basis of this characterization we present an algorithm to classify the simple maps from the plane to the plane and also give its implementation in computer algebra system SINGULAR.

Characterization of uni-modal parametric plane curve singularities

2017 ◽  
Vol 16 (02) ◽  
pp. 1750039
Author(s):  
Muhammad Ahsan Binyamin ◽  
Rabia ◽  
Hasan Mahmood ◽  
Junaid Alam Khan ◽  
Khawar Mehmood
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Plane Curve ◽  
Plane Curve Singularities ◽  
Parametric Plane ◽  
Curve Singularities

In this article we characterize the classification of uni-modal parametric plane curve singularities given by Ishikawa and Janeczko, in terms of invariants. On the basis of this characterization we present an algorithm to classify the uni-modal parametric plane curve singularities and also give its implementation in computer algebra system SINGULAR.

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.

scholarly journals Toral automorphisms and antiautomorphisms of rotation algebras

1999 ◽  
Vol 59 (2) ◽  
pp. 247-255
Author(s):  
Hu Yaohua ◽  
P.J. Stacey
Keyword(s):  
Complete Classification ◽  
Irrational Rotation ◽  
Complex Numbers ◽  
Toral Automorphisms

If U, V are the generators of a rational or irrational rotation C*-algebra then an automorphism φ of the algebra is determined by φ(U) = λUaVc and φ(V) = μUbVd where λ, μ are complex numbers of modulus 1 and a, b, c, d are integers with ad − bc = 1. If ad − bc = −1, then these formulae determine an antiautomorphsm of the algebra. The classification of such automorphisms and antiautomorphisms up to conjugacy by arbitrary automorphisms is studied and an almost complete classification is obtained.

scholarly journals SIMPLIFIED NUMERICAL FORM OF UNIVERSAL FINITE TYPE INVARIANT OF GAUSS WORDS

2013 ◽  
Vol 22 (08) ◽  
pp. 1350037
Author(s):  
TOMONORI FUKUNAGA ◽  
TAKAYUKI YAMAGUCHI ◽  
TAKAAKI YAMANOI
Keyword(s):  
Normal Form ◽  
Computer Program ◽  
Finite Type ◽  
Group Structure ◽  
Complete Classification ◽  
Finite Type Invariants ◽  
Partial Classification ◽  
Type Invariant

In this paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degrees 4, 5 and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.

scholarly journals Solving the Forward Position Problem of an In-Parallel Planar Manipulator in the Gauss Plane

2021 ◽  
Author(s):  
Sureyya Sahin
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Complex Numbers ◽  
Single Variable ◽  
Groebner Bases ◽  
Active Joint ◽  
Triangular Shape ◽  
Revolute Joints ◽  
Moving Platform ◽  
Position Problem

We study determining the posture of an in-parallel planar manipulator, which has three connectors composed of revolute, prismatic and revolute joints, from specified active joint variables. We construct an ideal in the field of complex numbers, and we introduce self inversive polynomials. We provide results for an in-parallel planar manipulator, which has a base and moving platform in right triangular shape. Using Sage computer algebra system, we compute its Groebner bases. We illustrate that the single variable polynomials obtained from the Groebner bases are self reciprocal.

scholarly journals Polarized Rigid Del Pezzo Surfaces in Low Codimension

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1567
Author(s):  
Muhammad Imran Qureshi
Keyword(s):  
Computer Algebra System ◽  
Symmetric Matrix ◽  
Single Equation ◽  
Complete Classification ◽  
Computer Assisted ◽  
Del Pezzo Surfaces ◽  
Linearly Independent ◽  
Del Pezzo ◽  
Detailed Computer

We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type ki×1ri(1,ai):3≤ri≤10,ki∈Z≥0 as well-formed and quasismooth varieties embedded in some weighted projective space. In particular, we present a collection of 147 such surfaces such that their image under their anti-canonical embeddings can be described by using one of the following sets of equations: a single equation, two linearly independent equations, five maximal Pfaffians of 5×5 skew symmetric matrix, and nine 2×2 minors of size 3 square matrix. This is a complete classification of such surfaces under certain carefully chosen bounds on the weights of ambient weighted projective spaces and it is largely based on detailed computer-assisted searches by using the computer algebra system MAGMA.

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