## What is Groebner?

Maple

What is Groebner? That was asked in different forms several times in MaplePrimes and MathStackExchange (for example, see http://math.stackexchange.com/questions/3550/using-gr?bner-bases-for-solving-polynomial-equations ). In view of this I think the presented post on Groebner basis will be useful. This post consists of two parts: its mathematical background and examples of solutions of polynomial systems by hand and with Maple.

Let us start. Up to Wiki http://en.wikipedia.org/wiki/Gr%C3%B6bner_basis ,Groebner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. This is implemented in Maple trough the Groebner package.
The simplest introduction to the topic I know is a well-written book of Ivan Arzhantsev (https://zbmath.org/?q=an:05864974) which includes the proofs of all the claimed theorems and the solutions of all the exercises. Here is its digest groebner.pdf done by me (The reader is assumed to be familiar with the ideal notion and ring notion (one may refresh her/his knowledge, looking in http://en.wikipedia.org/wiki/Ideal_%28ring_theory%29)). It should be noted that there is no easy reading about this serious matter.
Referring to the digest as appropriate, we solve the system
S:={a*b = c^2+c, a^2 =a+ b*c, a*c = b^2+b} by hand and with the Groebner package.
For the order a > b > c we construct its ideal
J(S):=<f1 = a*b-c^2-c,f2 = a^2-a-b*c, f3 = a*c-b^2-b>.
The link between f1 and f2 gives
f1*a-f2*b = (-c^2-c)*a + (a + b*c)*b = a*b -a*c + b^2*c -
a*c^2 =f4.
The reduction with f1 produces
f4 ->-a*c^2- a*c + b^2*c + c^2 +c =: f4.
Now the reduction with f3 produces
f4 -> -b^2- b - b*c +c^2 + c =:f4.
The link between f2 and f3 gives:
f2*c - f3*a = a*b +a*b^2 -a*c -b*c^2 = f5.
The reduction with f1 produces
f5 -> -a*c + c*b +c^2 +c =:f5.
The reduction with f3 produces
f5 -> -b^2 -b + c*b +c^2 +c =:f5.
The reduction with f4 produces
f5 -> 2b*c =: f5.
The link between f1 and f3
f1*c - f3*b = b^3 + b^2 -c^3 -c^2=:f6.
The reduction with f4 produces
f6 -> 2b*c + 2b*c^2 -2c^3 -2c^2=:f6.
At last, we reduce f6 by f5, obtaining f6:= -2c^3 -2c^2.
We see the minimal reduced Groebner basis of S consists of
a^2 -a -b*c, -b^2 -b- b*c +c^2 +c, -2c^3 - 2c^2.
Now we find the solution set of the system under consideration. The equation -2c^3 - 2c^2 = 0 implies
c=0, c=0, c=-1. The the equation -b^2 - b - b*c +c^2 + c = 0 gives
b = 0 , b = -1, b = 0, b = -1, b = 0, b = 0 respectively.
At last, knowing b and c, we find a from a^2 -a -b*c = 0.
Hence,
[{a = 0, b = 0, c = 0}, {a = 1, b = 0, c = 0}, {a = 0, b = -1, c = 0}], [{a = 0, b = 0, c = 0}, {a = 1, b = 0, c = 0}, {a = 0, b = -1, c = 0}], [{a = 0, b = 0, c = -1}].
The solution of the system under consideration by the Groebner package is somewhat different because Maple does not find the minimal reduced Groebner basis directly.

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## Integration Variables in Maple

by:

A customer on Twitter recently asked why Maple gives the following result:

The issue here is that the  in  is the same as the integration variable. 140 characters is not a lot to work with for a reply, so here is a longer explanation.

First, note that the process of integration treats the integration variable differently than the other variables, so that replacing another variable by the integration variable has a different effect depending on whether the replacement is done before or after the integration is performed. Consider this simple example:

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In other words, integration does not commute with substitution. This is a fundamental property of integration and in fact, Maple's eval has special rules to take this into account when you ask it to replace the integration variable.  For example, if you evaluate the inert form of the integral at , the substitution is performed explicitly:

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However, if you try to evaluate at , the evaluation is delayed until after the integral is evaluated:

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The eval command knows it shouldn't substitute into an integral when the substitution involves the variable of integration.

However, in the user's example, asking Maple for  is equivalent to substitution directly before the integration is performed, like this:

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which of course gives:

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Another way to have the two  variables be considered distinct is to explicitly make the integration variable a dummy by declaring it local:

Now the s are treated differently:

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Austin Roche

Senior Math Developer

Maplesoft

by:

Maple 18.02 contains improvements to many areas, including:

• Rendering of 2-D plots
• Help pages and examples
• Interactive components
• MATLAB 2014a support
• Math engine: Laplace transforms, complex floats, simplify
• Matrix import
• Typesetting
• Memory management
• OpenMaple API

To get this update, you can use Tools>Check for Updates from within Maple, or visit Maple 18.02 Downloads. (But depending on when you read this, Check for Updates may not have kicked in yet. If it doesn't find anything, wait until tomorrow morning and try again.)

• Efficiency improvements to the simulation engine, taking advantage of enhancements in Maple 18.02
• Improvements to C code generation for model export
• Improved handling of indexed variables when generating Modelica code

eithne

## I National Congress of Civil Engineering

Presentations of the first national congress of civil engineering developed at the University Cesar Vallejo. From 10 to 12 November 2014.

CONIC_UCV.pdf

(in spanish)

Lenin Araujo Castillo

Physics Pure

Computer Science

## November Live Webinars

by: Maple

Maplesoft regularly hosts live webinars on a variety of topics. Below you will find details on upcoming webinars we think may be of interest to the MaplePrimes community.  For the complete list of upcoming webinars, visit our website.

Creating Questions in Maple T.A. – Part #1

This webinar will demonstrate how to create questions in Maple T.A., Maplesoft’s testing and assessment solution for any course involving mathematics. The presentation will begin with an overview of the basic types of questions available, and then delve into how to create various types of questions in Maple T.A. Incorporating algorithms and feedback directly into questions will also be touched on. Finally, the session will wrap up with an explanation and several examples of how to create better questions using the question designer.

This first webinar in a two part series will cover true/false, multiple choice, numeric, mathematical formula, fill in the blank, sketch, and FBD questions. A second webinar that demonstrates more advanced question types will follow.

Clickable Calculus: Linear Algebra

In this webinar, Dr. Robert Lopez will apply the techniques of “Clickable Calculus” to standard calculations in Linear Algebra.

Clickable Calculus, the idea of powerful mathematics delivered using very visual, interactive point-and-click methods, offers educators a new generation of teaching and learning techniques. Clickable Calculus introduces a better way of engaging students so that they fully understand the materials they are being taught. It responds to the most common complaint of faculty who integrate software into the classroom – time is spent teaching the tool, not the concepts.

## Something to discuss about CASes

Maple

I'd like to pay attention to the recent article "The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?"

In particular, the authors consider the integral

int(abs(exp(2*Pi*Ix)+exp(2*Pi*I*y)),[x=0..1,y=0..1]),

stating "Both Mathematica and Maple return zero as the answer to this calculation. Yet this cannot be correct, because the integrand is clearly positive and nonzero in the indicated region". Unfortunately, they give only the Mathematica command to this end.

Of course, the integral under consideration is complicated so the the simple-minded trials

int(evalc(abs(exp((2*Pi*I)*x)+exp((2*Pi*I)*y))), [x = 0 .. 1, y = 0 .. 1]);

and

VectorCalculus:-int(evalc(abs(exp((2*Pi*I)*x)+exp((2*Pi*I)*y))), [x,y]=Rectangle( 0 .. 1, 0 .. 1));

fail. However,this can be found with Maple (I think with Mathematica too.) in such a way.

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## 2014 Maple T.A. User Summit - Day Two

by:

With the 2014 Maple T.A. User Summit completed and conference goers have returned to their home cities, it’s time to recap what happened on day 2. We started things off in the morning with an energized presentation by Prof. Jack Weiner of the University of Guelph. Jack pointed out that the University of Guelph has revolutionized their teaching and online homework solutions by using Maple and Maple T.A. He also gave an example of one of this famous Friday Specials – an example of math being used in the real world and discussed other teaching suggestions from his 40+ years of experience in the field.

## Physics Package Development

by: Maple 18

I figured I'd start a new thread for odd things I come across whilst using the new physics package.

I have found this, and am not sure if it is expected.

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## MapleNet 18

by: MapleNet

MapleNet 18 is now available.  MapleNet 18 provides increased mobile support and eliminates the need for a Java plug-in when interacting with Maple documents in a web browser. See What’s New in MapleNet 18 for more information.

eithne

## Visualization of meme propagation on the internet

by:

Hi, we recently put together a web video on how memes spread on the internet using several visualizations generated from Maple 18:

http://youtu.be/vEhAkEPwESI

Found the new ability to specify a background image for plots to be very helpful.

## 2014 Maple T.A. User Summit - Day One

by:

Reporting from Amsterdam, it's a pleasure to report on day one of the 2014 Maple T.A. User Summit. Being our first Maple T.A. User Summit, we wanted to ensure attendees were not only given the opportunity to sit-in on key note presentations from various university or college professors, high school teachers and Maplesoft staff, but to also engage in active discussions with each other on how they have implemented Maple T.A. at their institution.

We started things off by hearing an encouraging talk by Maplesoft’s president and CEO Jim Cooper. Jim started things off with a question to get everyone thinking; “How will someone born today be educated in the 2030s?” From there, we heard about Maplesoft’s vision on education, learning, and questions we have to ask ourselves today to be prepared for the future.

Up next was Louise Krmpotic, Director of Business Development. Louise discussed content and Maple T.A. This included an overview of our content team operations, what content is currently available today, and how users can engage themselves in the Maple T.A. community and get involved in sharing their own content with other users.

We then heard our first keynote presentation by Professor Steve Furino and Rachael Vanbruggen of the University of Waterloo.  We were provided with a brief history of the University of Waterloo and mathematics as well as their ever expanding initiative in brining math courses into an online environment both at the university level and high school level. We then heard in detail of how Maplesoft technologies have been implemented in various math courses and the successes and challenges of creating their own content.

I (Jonny Zivku, Product Manager of Maple T.A.) then delivered a presentation on all the new features in Maple T.A. 10. I won’t get into detail about the new features in this post, but if you’d like to read more about it, check out my previous post from a few weeks ago.

Meta Keijzer-de Ruijter of TU Delft University then took the floor and delivered our second keynote presentation. She discussed the history of Delft as well as the new initiative, the Delft Extension School. She then went on to discuss Delft’s experiences with implementing Maple T.A. at their campus and maintaining it since 2007 as well as how they’ve managed to maintain their academic integrity while using online tools. We also had the opportunity of seeing several examples of some of their excellent questions they’ve created which included adaptive, math apps, algorithms, maple-graded and more.

After a delicious lunch break, Paul DeMarco from Maplesoft, Director of Maple and Maple T.A. Development, talked about the future of testing and assessment. Paul went over various topics and how we envision them changing which included partial marks, skills assessment, learning, feedback, and content.

Jonathan Kress from the University of New South Wales was up next and discussed their experiences with implementing Maple T.A. into their mathematics and statistics courses at a first year, second year, and higher level of learning. He then discussed the various scenarios for how Maple T.A. is deployed which included both formative and summative testing. Moving on, we then were briefed on Maple T.A. use from a student's perspective and an overview of various pieces of content.

We then moved on to an engaging panel discussion which featured Grahame Smart, math and e-learning consultant, Professor Marina Marchisio of the University of Turin and Dr. Alice Barana also from the University of Turin. Grahame first started things off by discussing how he doubled the pass rates in his prevoius high school using investigative and interactive learning with Maple T.A. Marina and Dr. Barana then gave us a brief overview of Maple T.A. at the University of Turin and their exciting PP&S project. The panel then answered various questions from the audience

William Rybolt of Babson College then closed off the presentations with a discussion about how his school has been a long time user of EDU, Maple T.A.’s predecessor.  Going from ungraded web pages, web forms, and Excel, we heard about Babson’s attempts at converting paper-based assignments into an online format until 2003 when they decided to adopt EDU.

To end the day, we enjoyed a nice cruise on the canals of Amsterdam while enjoying a delicious three course meal. Not a bad way to end the day!

Jonny
Maplesoft Product Manager, Maple T.A.

## MapleSim 7 is now available

We have just released a new version of MapleSim.

MapleSim 7 makes it substantially easier to explore and validate designs, create and manage libraries of custom components, and use your MapleSim models with other tools. It includes:

• Easy model investigation. A new Results Manager gives you greater flexibility when it comes to investigating your simulation results, including the ability to compare simulation runs on the same axes, instantly plot both probed and unprobed variables, and easily create custom plots.
• Convenient library creation. With MapleSim 7, it is significantly easier to create, manage, and share libraries of custom components.
• Improved Modelica support. MapleSim 7 expands the support of the Modelica language so that more Modelica definitions can be used directly inside MapleSim.

We have also updated and expanded the MapleSim 7 family of add-on products:

• The new MapleSim Battery Library, which is available as a separate add-on, allows you to incorporate physics-based predictive models of battery cells into your system models so you can take battery behavior into account early in the design process.
• The MapleSim Connector for FMI, which allows engineers to share very efficient, high-fidelity models created in MapleSim with other modeling tools, has been expanded to support more export formats for co-simulation and model exchange.

eithne

## Computer Algebra for Theoretical Physics

by:

Last week the Physics package was presented in a talk at the Perimeter Institute for Theoretical Physics and in a combined Applied Mathematics and Physics Seminar at the University of Waterloo. The presentation at the Perimeter Institute got recorded. It was a nice opportunity to surprise people with the recent advances in the package. It follows the presentation with sections closed, and at the end there is a link to a pdf with the sections open and to the related worksheet, used to run the computations in real time during the presentation.

COMPUTER ALGEBRA FOR THEORETICAL PHYSICS

 Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing on a Computer Algebra worksheet. On the other hand, recent developments in the Maple system implemented most of the mathematical objects and mathematics used in theoretical physics computations, and dramatically approximated the notation used in the computer to the one used in paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum. In connection, in this talk the Physics project at Maplesoft is presented and the resulting Physics package illustrated tackling problems in classical and quantum mechanics, general relativity and field theory. In addition to the 10 a.m lecture, there will be a hands-on workshop at 1pm in the Alice Room.

 ... Why computers? We can concentrate more on the ideas instead of on the algebraic manipulations   We can extend results with ease   We can explore the mathematics surrounding a problem   We can share results in a reproducible way

Representation issues that were preventing the use of computer algebra in Physics

Notation and related mathematical methods that were missing:

coordinate free representations for vectors and vectorial differential operators,

covariant tensors distinguished from contravariant tensors,

functional differentiation, relativity differential operators and sum rule for tensor contracted (repeated) indices

Bras, Kets, projectors and all related to Dirac's notation in Quantum Mechanics

Inert representations of operations, mathematical functions, and related typesetting were missing:

inert versus active representations for mathematical operations

ability to move from inert to active representations of computations and viceversa as necessary

hand-like style for entering computations and texbook-like notation for displaying results

Key elements of the computational domain of theoretical physics were missing:

ability to handle products and derivatives involving commutative, anticommutative and noncommutative variables and functions

ability to perform computations taking into account custom-defined algebra rules of different kinds

(problem related commutator, anticommutator, bracket, etc. rules)

 Vector and tensor notation in mechanics, electrodynamics and relativity
 Dirac's notation in quantum mechanics

 • Computer algebra systems were not originally designed to work with this compact notation, having attached so dense mathematical contents, active and inert representations of operations, not commutative and customizable algebraic computational domain, and the related mathematical methods, all this typically present in computations in theoretical physics.
 • This situation has changed. The notation and related mathematical methods are now implemented.

Tackling examples with the Physics package

Classical Mechanics

Inertia tensor for a triatomic molecule

Problem: Determine the Inertia tensor of a triatomic molecule that has the form of an isosceles triangle with two masses  in the extremes of the base and mass  in the third vertex. The distance between the two masses  is equal to a, and the height of the triangle is equal to h.

 Solution

Quantum mechanics

Quantization of the energy of a particle in a magnetic field

Show that the energy of a particle in a constant magnetic field oriented along the z axis can be written as

where and  are creation and anihilation operators.

 Solution
 The quantum operator components of  satisfy

Unitary Operators in Quantum Mechanics

(with Pascal Szriftgiser, from Laboratoire PhLAM, Université Lille 1, France)

A linear operator U is unitary if , in which case, Unitary operators are used to change the basis inside an Hilbert space, which physically means changing the point of view of the considered problem, but not the underlying physics. Examples: translations, rotations and the parity operator.

 1) Eigenvalues of an unitary operator and exponential of Hermitian operators
 2) Properties of unitary operators
 3) Schrödinger equation and unitary transform
 4) Translation operators

Classical Field Theory

The field equations for a quantum system of identical particles

Problem: derive the field equation describing the ground state of a quantum system of identical particles (bosons), that is, the Gross-Pitaevskii equation (GPE). This equation is particularly useful to describe Bose-Einstein condensates (BEC).

 Solution
 The field equations for the  model
 Maxwell equations departing from the 4-dimensional Action for Electrodynamics

General Relativity

Given the spacetime metric,

a) Compute the trace of

where  is some function of the radial coordinate,  is the Ricci tensor,  is the covariant derivative operator and  is the stress-energy tensor

b) Compute the components of the traceless part of   of item a)

c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of   obtained in b)

Background: paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by P. Fiziev.

 a) The trace of
 b) The components of the traceless part of
 c) An exact solution for the nonlinear system of differential equations conformed by the components of
 The Physics Project "Physics" is a software project at Maplesoft that started in 2006. The idea is to develop a computational symbolic/numeric environment specifically for Physics, targeting educational and research needs in equal footing, and resembling as much as possible the flexible style of computations used with paper and pencil. The main reference for the project is the Landau and Lifshitz Course of Theoretical Physics.   A first version of "Physics" with basic functionality appeared in 2007. Since then the package has been growing every year, including now, among other things, a searcheable database of solutions to Einstein equations and a new dedicated programming language for Physics.   Since August/2013, weekly updates of the Physics package are distributed on the web, including the new developments related to our plan as well as related to people's feedback.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

## 2014 Maple T.A. User Summit – Schedule

by:

As previously announced, Maplesoft will be hosting  the 2014 Maple T.A. User Summit this October 22 – 24 in Amsterdam, The Netherlands. You might have heard about the launch of Maple T.A. 10. The User Summit in Amsterdam is a perfect opportunity to get to know more, see the new features in action, and meet Maple T.A. users from around the world.

We are happy to announce that the schedule has been finalized! The event will feature keynote and user presentations by prominent educators from around the world, first-hand discussions by Maplesoft representatives, exciting social events, and training sessions.

As you can see, this event has shaped up to be a very exciting summit for Maple T.A. users. After seeing this schedule you may be wondering why you didn’t sign-up – don’t worry, it’s not too late! To register, please visit our website: https://webstore.maplesoft.com/taconference/register.aspx

I hope to see you there!

Jonny
Maplesoft Product Manager, Maple T.A.

## Solutions of Differential Equations with Embedded...

Maple 18

The Embedded Components are containers that currently use industries for modeling complex systems to find viable solutions in real time and thus avoid huge wait times and overload our computer; by this paper should show you how to implement a dynamic worksheet through Embedded Components in Maple; it goes from finding solutions to ordinary differential equations partial; which interact with the researcher using different parameters.
Using graphical programming will find immediate solutions to selected problems in science and engineering criteria of variability and boundary conditions evolving development with buttons on multiple actions.

cimac_2014.pdf

(in spanish)

Solutions_of_Differential_Equations_with_Embedded_Components.mw

Lenin Araujo Castillo

Physics Pure

Computer Science

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