Abstract Algebra: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=131
en-us2019 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 14 Nov 2019 18:05:34 GMTThu, 14 Nov 2019 18:05:34 GMTNew applications in the Abstract Algebra categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgAbstract Algebra: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=131
Non-Nested Real Algebraic Numbers By Maple
https://www.maplesoft.com/applications/view.aspx?SID=154578&ref=Feed
Procedures are introduced to represent non-nested real algebraic numbers by matrices and compute conjugates and minimal polynomials of these numbers.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Non-Nested Real Algebraic Numbers By Maple" style="max-width: 25%;" align="left"/>Procedures are introduced to represent non-nested real algebraic numbers by matrices and compute conjugates and minimal polynomials of these numbers.https://www.maplesoft.com/applications/view.aspx?SID=154578&ref=FeedSat, 26 Oct 2019 04:00:00 ZKahtan H. AlzubaidyKahtan H. AlzubaidyInvariant Theory of Finite Groups by Maple
https://www.maplesoft.com/applications/view.aspx?SID=154378&ref=Feed
This article deals with the computations of the generators and relations of the invariant polynomial ring F[x,y]^G, where G is a finite group of 2-square matrices over F acting linearly on F[x,y].F is a field of characteristic 0.The group G is of low order.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Invariant Theory of Finite Groups by Maple" style="max-width: 25%;" align="left"/>This article deals with the computations of the generators and relations of the invariant polynomial ring F[x,y]^G, where G is a finite group of 2-square matrices over F acting linearly on F[x,y].F is a field of characteristic 0.The group G is of low order.https://www.maplesoft.com/applications/view.aspx?SID=154378&ref=FeedTue, 19 Dec 2017 05:00:00 ZKahtan H. AlzubaidyKahtan H. AlzubaidyFibonacci Numbers
https://www.maplesoft.com/applications/view.aspx?SID=154362&ref=Feed
Many programming language tutorials have an example about computing Fibonacci numbers to illustrate recursion. Usually, however, these simple examples exhibit an abysmal runtime behaviour, namely, exponential in the index.
<BR><BR>
In this presentation, several more efficient ways of computing Fibonacci numbers, using Maple, are discussed. The best algorithm presented is based on doubling formulae for the Fibonacci numbers, which we also prove using Maple.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Fibonacci Numbers" style="max-width: 25%;" align="left"/>Many programming language tutorials have an example about computing Fibonacci numbers to illustrate recursion. Usually, however, these simple examples exhibit an abysmal runtime behaviour, namely, exponential in the index.
<BR><BR>
In this presentation, several more efficient ways of computing Fibonacci numbers, using Maple, are discussed. The best algorithm presented is based on doubling formulae for the Fibonacci numbers, which we also prove using Maple.https://www.maplesoft.com/applications/view.aspx?SID=154362&ref=FeedTue, 21 Nov 2017 05:00:00 ZDr. Jürgen GerhardDr. Jürgen GerhardQUADRATICINT - A MAPLE PACKAGE FOR WORKING WITH ELEMENTS OF Z[sqrt(d)]
https://www.maplesoft.com/applications/view.aspx?SID=154235&ref=Feed
This paper presents the description of the package for working with elements of
Z[sqrt(d)]
where
d <> 1
is a squarefree integer. Given in the paper are the program codes of package commands and examples. Developed software will be useful for specialists in the field of abstract algebra and applications.
Key words: Computer algebra, symbolic computation, algebraic number theory, ring theory, rings of integers of algebraic number fields, rings
Z[sqrt(d)]
.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="QUADRATICINT - A MAPLE PACKAGE FOR WORKING WITH ELEMENTS OF Z[sqrt(d)]" style="max-width: 25%;" align="left"/>This paper presents the description of the package for working with elements of
Z[sqrt(d)]
where
d <> 1
is a squarefree integer. Given in the paper are the program codes of package commands and examples. Developed software will be useful for specialists in the field of abstract algebra and applications.
Key words: Computer algebra, symbolic computation, algebraic number theory, ring theory, rings of integers of algebraic number fields, rings
Z[sqrt(d)]
.https://www.maplesoft.com/applications/view.aspx?SID=154235&ref=FeedTue, 02 May 2017 04:00:00 ZNatalia TsybulskaNatalia TsybulskaAlternating Polynomials by Maple
https://www.maplesoft.com/applications/view.aspx?SID=154233&ref=Feed
This application contains few procedures to do computations on alternating and symmetric polynomials. They are based on the fundamental theorem of symmetric polynomials and on the fundamental theorem of alternating polynomials by using Groebner’s bases.
The procedures are working very well, when the coefficients are numerical.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Alternating Polynomials by Maple" style="max-width: 25%;" align="left"/>This application contains few procedures to do computations on alternating and symmetric polynomials. They are based on the fundamental theorem of symmetric polynomials and on the fundamental theorem of alternating polynomials by using Groebner’s bases.
The procedures are working very well, when the coefficients are numerical.https://www.maplesoft.com/applications/view.aspx?SID=154233&ref=FeedFri, 07 Apr 2017 04:00:00 ZKahtan H. AlzubaidyKahtan H. AlzubaidyMetacyclic Groups by Maple
https://www.maplesoft.com/applications/view.aspx?SID=153918&ref=Feed
A class of metacyclic groups is considered for Maple computations. Tables of groups, inverse elements and orders are found out. Centers and conjugate classes of these groups are computed also. The method used is based on construction of a complex copy of the metacyclic group to make Maple computations easier.
The advantage of this treatment of metacyclic groups is that mathematical form of the subject is maintained to a great degree.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Metacyclic Groups by Maple" style="max-width: 25%;" align="left"/>A class of metacyclic groups is considered for Maple computations. Tables of groups, inverse elements and orders are found out. Centers and conjugate classes of these groups are computed also. The method used is based on construction of a complex copy of the metacyclic group to make Maple computations easier.
The advantage of this treatment of metacyclic groups is that mathematical form of the subject is maintained to a great degree.https://www.maplesoft.com/applications/view.aspx?SID=153918&ref=FeedTue, 10 Nov 2015 05:00:00 ZKahtan H. AlzubaidyKahtan H. AlzubaidyMultiplicative Cyclic Groups by Maple
https://www.maplesoft.com/applications/view.aspx?SID=153897&ref=Feed
<P>In this article we introduce cyclic groups in multiplicative notations by using Maple 13. Group table,order table,and inverse table are given.</P>
<P>
All generators and all subgroups of a multiplicative cyclic group are given also. Hasse diagrams of subgroup lattices are shown. All homomorphisms and automorphisms of such groups are computed as well as the kernel and image of a homomorphism.</P><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Multiplicative Cyclic Groups by Maple" style="max-width: 25%;" align="left"/><P>In this article we introduce cyclic groups in multiplicative notations by using Maple 13. Group table,order table,and inverse table are given.</P>
<P>
All generators and all subgroups of a multiplicative cyclic group are given also. Hasse diagrams of subgroup lattices are shown. All homomorphisms and automorphisms of such groups are computed as well as the kernel and image of a homomorphism.</P>https://www.maplesoft.com/applications/view.aspx?SID=153897&ref=FeedThu, 15 Oct 2015 04:00:00 ZKahtan H. AlzubaidyKahtan H. AlzubaidyQuotient Polynomial Rings by Maple
https://www.maplesoft.com/applications/view.aspx?SID=153872&ref=Feed
Quotient polynomial rings over the infinite field containing ℚ are involved. The computations concern the univariate polynomial rings and the multivariate polynomial rings in two variables. In both cases a vector space basis for the quotient is constructed. The case of of several variables includes cases of infinite dimensions.We shall restrict ourselves to the finite computations. Ring operations of addition and multiplication on the quotients are computed as well.
Goebner basis is used and the computations are carried out in Maple 13.<img src="https://www.maplesoft.com/view.aspx?si=153872/0038a244bd8b097c72d4ef733ddf7f8c.gif" alt="Quotient Polynomial Rings by Maple" style="max-width: 25%;" align="left"/>Quotient polynomial rings over the infinite field containing ℚ are involved. The computations concern the univariate polynomial rings and the multivariate polynomial rings in two variables. In both cases a vector space basis for the quotient is constructed. The case of of several variables includes cases of infinite dimensions.We shall restrict ourselves to the finite computations. Ring operations of addition and multiplication on the quotients are computed as well.
Goebner basis is used and the computations are carried out in Maple 13.https://www.maplesoft.com/applications/view.aspx?SID=153872&ref=FeedSat, 12 Sep 2015 04:00:00 ZKahtan H. AlzubaidyKahtan H. AlzubaidyTips and Techniques: Working with Finitely Presented Groups in Maple
https://www.maplesoft.com/applications/view.aspx?SID=153852&ref=Feed
This Tips and Techniques article introduces Maple's facilities for working with finitely presented groups. A finitely presented group is a group defined by means of a finite number of generators, and a finite number of defining relations. It is one of the principal ways in which a group may be represented on the computer, and is virtually the only representation that effectively allows us to compute with many infinite groups.<img src="https://www.maplesoft.com/view.aspx?si=153852/thumb.jpg" alt="Tips and Techniques: Working with Finitely Presented Groups in Maple" style="max-width: 25%;" align="left"/>This Tips and Techniques article introduces Maple's facilities for working with finitely presented groups. A finitely presented group is a group defined by means of a finite number of generators, and a finite number of defining relations. It is one of the principal ways in which a group may be represented on the computer, and is virtually the only representation that effectively allows us to compute with many infinite groups.https://www.maplesoft.com/applications/view.aspx?SID=153852&ref=FeedTue, 25 Aug 2015 04:00:00 ZMaplesoftMaplesoftSymmetric Polynomials by Maple
https://www.maplesoft.com/applications/view.aspx?SID=153837&ref=Feed
<p>Procedures are presented in Maple 13 to make computations in symmetric polynomials by using Groebner basis.A simple application to the theory of equations is given.</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Symmetric Polynomials by Maple" style="max-width: 25%;" align="left"/><p>Procedures are presented in Maple 13 to make computations in symmetric polynomials by using Groebner basis.A simple application to the theory of equations is given.</p>https://www.maplesoft.com/applications/view.aspx?SID=153837&ref=FeedThu, 06 Aug 2015 04:00:00 ZKahtan H. AlzubaidyKahtan H. AlzubaidyGroebner Bases: What are They and What are They Useful For?
https://www.maplesoft.com/applications/view.aspx?SID=153693&ref=Feed
Since they were first introduced in 1965, Groebner bases have proven to be an invaluable contribution to mathematics and computer science. All general purpose computer algebra systems like Maple have Groebner basis implementations. But what is a Groebner basis? And what applications do Groebner bases have? In this Tips and Techniques article, I’ll give some examples of the main application of Groebner bases, which is to solve systems of polynomial equations.<img src="https://www.maplesoft.com/view.aspx?si=153693/thumb.jpg" alt="Groebner Bases: What are They and What are They Useful For?" style="max-width: 25%;" align="left"/>Since they were first introduced in 1965, Groebner bases have proven to be an invaluable contribution to mathematics and computer science. All general purpose computer algebra systems like Maple have Groebner basis implementations. But what is a Groebner basis? And what applications do Groebner bases have? In this Tips and Techniques article, I’ll give some examples of the main application of Groebner bases, which is to solve systems of polynomial equations.https://www.maplesoft.com/applications/view.aspx?SID=153693&ref=FeedFri, 17 Oct 2014 04:00:00 ZProf. Michael MonaganProf. Michael MonaganIndependenceModel package
https://www.maplesoft.com/applications/view.aspx?SID=148816&ref=Feed
<p>The main purpose of this work was to write a procedure to implement an algorithm based on the Diaconis Sturmfels algorithm to compute the Monte Carlo p-value of the independence model considered, but we present a package containing also some preliminary commands that can be useful to everyone studying an independence model.</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="IndependenceModel package" style="max-width: 25%;" align="left"/><p>The main purpose of this work was to write a procedure to implement an algorithm based on the Diaconis Sturmfels algorithm to compute the Monte Carlo p-value of the independence model considered, but we present a package containing also some preliminary commands that can be useful to everyone studying an independence model.</p>https://www.maplesoft.com/applications/view.aspx?SID=148816&ref=FeedTue, 25 Jun 2013 04:00:00 ZValentina TrioloValentina TrioloA new algorithm for computing moments of complex non-central Wishart distributions
https://www.maplesoft.com/applications/view.aspx?SID=143890&ref=Feed
<p>A new algorithm for computing joint moments of complex non-central Wishart distributions <em>W </em>is provided, relied on a symbolic method which is particularly suited to be implemented for multivariate statistical distributions by using necklaces.</p><img src="https://www.maplesoft.com/view.aspx?si=143890/logo.jpg" alt="A new algorithm for computing moments of complex non-central Wishart distributions" style="max-width: 25%;" align="left"/><p>A new algorithm for computing joint moments of complex non-central Wishart distributions <em>W </em>is provided, relied on a symbolic method which is particularly suited to be implemented for multivariate statistical distributions by using necklaces.</p>https://www.maplesoft.com/applications/view.aspx?SID=143890&ref=FeedMon, 25 Feb 2013 05:00:00 ZDr. Giuseppe GuarinoDr. Giuseppe GuarinoPolynomial System Solving in Maple 16
https://www.maplesoft.com/applications/view.aspx?SID=132208&ref=Feed
Computing and manipulating the real solutions of a polynomial system is a requirement for many application areas, such as biological modeling, robotics, program verification, and control design, to name just a few. For example, an important problem in computational biology is to study the stability of the equilibria (or steady states) of biological systems. This question can often be reduced to solving a parametric system of polynomial equations and inequalities. In this application, these techniques are used to perform stability analysis of a parametric dynamical system and verify mathematical identities through branch cut analysis.<img src="https://www.maplesoft.com/view.aspx?si=132208/thumb.jpg" alt="Polynomial System Solving in Maple 16" style="max-width: 25%;" align="left"/>Computing and manipulating the real solutions of a polynomial system is a requirement for many application areas, such as biological modeling, robotics, program verification, and control design, to name just a few. For example, an important problem in computational biology is to study the stability of the equilibria (or steady states) of biological systems. This question can often be reduced to solving a parametric system of polynomial equations and inequalities. In this application, these techniques are used to perform stability analysis of a parametric dynamical system and verify mathematical identities through branch cut analysis.https://www.maplesoft.com/applications/view.aspx?SID=132208&ref=FeedTue, 27 Mar 2012 04:00:00 ZMaplesoftMaplesoftThe Origin of Complex Numbers
https://www.maplesoft.com/applications/view.aspx?SID=126618&ref=Feed
The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="The Origin of Complex Numbers" style="max-width: 25%;" align="left"/>The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.https://www.maplesoft.com/applications/view.aspx?SID=126618&ref=FeedFri, 14 Oct 2011 04:00:00 ZDr. John MathewsDr. John MathewsThe Advanced Encryption Standard and its modes of operation
https://www.maplesoft.com/applications/view.aspx?SID=6618&ref=Feed
<p>This is an update, labeled version 1.1, to the existing application The Advanced Encryption Standard and its modes of operation.</p>
<p>Version 1.1: Key generation function and related functions updated to facilitate the use of externally generated seeds. Some minor changes to presentation.</p>
<p>Version 1.0: Implementation of encryption and authentication schemes that use the Advanced Encryption Standard (AES) as their underlying block cipher. These schemes are constructed by using all the modes of operation for block ciphers so far approved by NIST (the US National Institute of Standards of Technology), namely, the five confidentiality modes: ECB, CBC, CFB, OFB and CTR, the authentication mode CMAC, and the "authenticated encryption" modes CCM and GCM/GMAC. The implementation is able to encrypt/decrypt and/or authenticate messages in several formats, including binary files, and we use it to explore the basic properties of these schemes. The implementation contains also detailed explanations of all the procedures used, including the lower level ones, and discusses both the programming and the cryptographic aspects involved.</p><img src="https://www.maplesoft.com/view.aspx?si=6618/AES_1608.gif" alt="The Advanced Encryption Standard and its modes of operation" style="max-width: 25%;" align="left"/><p>This is an update, labeled version 1.1, to the existing application The Advanced Encryption Standard and its modes of operation.</p>
<p>Version 1.1: Key generation function and related functions updated to facilitate the use of externally generated seeds. Some minor changes to presentation.</p>
<p>Version 1.0: Implementation of encryption and authentication schemes that use the Advanced Encryption Standard (AES) as their underlying block cipher. These schemes are constructed by using all the modes of operation for block ciphers so far approved by NIST (the US National Institute of Standards of Technology), namely, the five confidentiality modes: ECB, CBC, CFB, OFB and CTR, the authentication mode CMAC, and the "authenticated encryption" modes CCM and GCM/GMAC. The implementation is able to encrypt/decrypt and/or authenticate messages in several formats, including binary files, and we use it to explore the basic properties of these schemes. The implementation contains also detailed explanations of all the procedures used, including the lower level ones, and discusses both the programming and the cryptographic aspects involved.</p>https://www.maplesoft.com/applications/view.aspx?SID=6618&ref=FeedMon, 20 Jun 2011 04:00:00 ZJosé Luis Gómez PardoJosé Luis Gómez PardoQuaternions
https://www.maplesoft.com/applications/view.aspx?SID=96897&ref=Feed
<p>Quaternions for Maple™ is an easy-to-use toolbox for Maple which allows professionals, researchers, and students to learn about, experiment with, and model systems through quaternions in the Maple worksheet environment. It transforms Maple into a system which works with quaternions as seamlessly as the basic system deals with complex numbers. The functionality covers many of the basic principles of quaternions including an environment where the multiplication operator (*) is no longer commutative.</p><img src="https://www.maplesoft.com/view.aspx?si=96897/quaternions_logo_s.jpg" alt="Quaternions" style="max-width: 25%;" align="left"/><p>Quaternions for Maple™ is an easy-to-use toolbox for Maple which allows professionals, researchers, and students to learn about, experiment with, and model systems through quaternions in the Maple worksheet environment. It transforms Maple into a system which works with quaternions as seamlessly as the basic system deals with complex numbers. The functionality covers many of the basic principles of quaternions including an environment where the multiplication operator (*) is no longer commutative.</p>https://www.maplesoft.com/applications/view.aspx?SID=96897&ref=FeedWed, 15 Sep 2010 04:00:00 ZDouglas HarderDouglas HarderA new approach to Sheppard’s corrections
https://www.maplesoft.com/applications/view.aspx?SID=59515&ref=Feed
<p>In the real world, variables are observed and recorded in finite precision through a rounding or coarsening operation, i.e. a grouping rule. Grouping includes also censoring or splitting data into categories during collection or publication, and so it involves continuous as well as discrete parent distributions. Sheppard's corrections are formulae which improve the computation of moments when data are grouped into classes. Here we give two speed procedures "raw2grp" and "grp2raw" to compute the correction to raw moments in terms of grouped moments (and viceversa) both in the continuous case and in the discrete case and both for univariate and multivariate parent distributions.</p><img src="https://www.maplesoft.com/view.aspx?si=59515/0\images\SceppardCorrectio_1.gif" alt="A new approach to Sheppard’s corrections" style="max-width: 25%;" align="left"/><p>In the real world, variables are observed and recorded in finite precision through a rounding or coarsening operation, i.e. a grouping rule. Grouping includes also censoring or splitting data into categories during collection or publication, and so it involves continuous as well as discrete parent distributions. Sheppard's corrections are formulae which improve the computation of moments when data are grouped into classes. Here we give two speed procedures "raw2grp" and "grp2raw" to compute the correction to raw moments in terms of grouped moments (and viceversa) both in the continuous case and in the discrete case and both for univariate and multivariate parent distributions.</p>https://www.maplesoft.com/applications/view.aspx?SID=59515&ref=FeedFri, 30 Apr 2010 04:00:00 ZDr. Giuseppe GuarinoDr. Giuseppe GuarinoThe CayleyDickson Algebra from 4D to 256D
https://www.maplesoft.com/applications/view.aspx?SID=35420&ref=Feed
<p>There are higher dimensional numbers besides complex numbers. There are also hypercomplex numbers, such as, quaternions (4 D), octonions (8 D), sedenions (16 D), pathions (32 D), chingons (64 D), routons (128 D), voudons (256 D), and so on, without end. These names were coined by Robert P.C. de Marrais and Tony Smith. It is an alternate naming system providing relief from the difficult Latin names, such as:<br /> trigintaduonions (32 D), sexagintaquatronions (64 D), centumduodetrigintanions (128 D), and ducentiquinquagintasexions (256 D).</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="The CayleyDickson Algebra from 4D to 256D" style="max-width: 25%;" align="left"/><p>There are higher dimensional numbers besides complex numbers. There are also hypercomplex numbers, such as, quaternions (4 D), octonions (8 D), sedenions (16 D), pathions (32 D), chingons (64 D), routons (128 D), voudons (256 D), and so on, without end. These names were coined by Robert P.C. de Marrais and Tony Smith. It is an alternate naming system providing relief from the difficult Latin names, such as:<br /> trigintaduonions (32 D), sexagintaquatronions (64 D), centumduodetrigintanions (128 D), and ducentiquinquagintasexions (256 D).</p>https://www.maplesoft.com/applications/view.aspx?SID=35420&ref=FeedFri, 23 Apr 2010 04:00:00 ZMichael CarterMichael CarterQuaternions, Octonions and Sedenions
https://www.maplesoft.com/applications/view.aspx?SID=35196&ref=Feed
<p>This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Quaternions, Octonions and Sedenions" style="max-width: 25%;" align="left"/><p>This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.</p>https://www.maplesoft.com/applications/view.aspx?SID=35196&ref=FeedFri, 16 Apr 2010 04:00:00 ZDr. Michael Angel Carter
Dr. Michael Angel Carter