The following example arrived in my email inbox a few weeks ago. It spurred a short but lively thread of discussion amongst some Maple developers.

I thought that it was interesting enough to post here. I'll hold off on giving my own opinion right away, because I'm curious to read what other MaplePrimes members might write about it.

> q := (6*((1/3)*a-1/9))/(36*a-116+12*sqrt(12*a^3-3*a^2-54*a+93))^(1/3);
                                   6 (a/3 - 1/9...

Some time ago I was asked the question: do you know how to do a change of variables in a multi-dimensional definite integral?  I thought I knew, but I was wrong.  I only know how to do a change of variable in a multi-dimensional indefinite integral. 

This question really intrigued me.  Of course, this is such an obvious question, I just started pulling out textbook after textbook off my bookshelf, expecting to find the answer...

I recently submitted my work to Maple Application Center and I received a bug report from a staff. Then, I resubmitted it after fixing bugs. However, I have a bug report again (^_^; Yes, this is because my work was poor, but in other words, all applications in Maple Application center that passed strict check by staff are all guaranteed to have good quality. I am sure that everyone can find good tools for education and research. We should utilize them. If we can not find applications that we want, let us develop works and submit them ! Yasuyuki Nakamura

On his blog, Jaime Zawinski (of Netscape and XEmacs fame) relates a tale of finding limits in the (supposedly) unlimited big number representation on a TI Lisp machine in the early 1990s. It is an amusing story, and it makes me wonder if GnuMP is has a similar limit on a different scale.  Or in other words, is there a positive integer small enough to fit into memory  (assuming 64 bit address space) but that cannot actually be constructed in GnuMP due to limits in the implementation? Does someone here know enough about the GnuMP internals to give the answer?

A search I was doing dug up this old gem, involving a discussion between Gerald Edgar and I over a Maple problem 16 years ago!

Easy challenge: improve on my solution to Gerald's problem.

History challenge: my email address shows as wmsical!jjcarett@watmath.waterloo.edu.  Can you puzzle that out?  That is really two questions, a) how is that an email address and b) what is 'wmsical' ?

The recent thread A crossprod problem, although not directly related, inspired me to write this blog entry about bilinear cross products and their noticable property [see for instance P. Lounesto, Clifford Algebras and Spinors, 2nd ed. (Cambridge University Press, Cambridge, 2001)]:

Theorem: A bilinear cross product obeying the orthogonality property and the Pythagorean property, see below, exists only in 3 or 7 (real or complex) dimensions.

As many here know, Gaston Gonnet is a co-founder of Maple.  SIAM, in its History of Numerical Analysis and Scientific Computing project, has now published a very long interview with Gaston.  For those who like a good yarn, as well as details of the history of Maple and Maplesoft, it makes for fascinating reading.

My favourite quotes are:

Inspired by Jacques' blog entry Introduction to transseries, concerning a paper by Gerald A. Edgar, using Maple and published at arXiv, I here take the liberty to refer to a recent paper of mine which also uses Maple and is published at arXiv. The link is:

Linking electroweak and gravitational generators.

Most probably, this paper would not have existed without the possibility of performing lots of calculations in Maple, using for instance my own package COSVAM which deals with the octonions, the largest division algebra over the reals.

For instance, the pivotal Eq. (5) of the paper would probably not have been discovered by me using pen and paper. It was accidentally discovered while performing some Maple calculations with a different objective in mind.

Second note added: The issue below seems to have been resolved by clearing the cache of my Firefox browser, i.e., it seems to have been purely a local problem.

Inspired by the blog post Find a point in every region defined by a system of linear equations, I have come up with the following method to find a point inside each bounded region. The assumptions are:

• No two lines are parallel.
• No three lines are coincident.

Due to numerical instability, it seems, using floats, the coefficients of the equations of the lines are taken to be integers (they could also have been taken to be fractions, of course). Then the method goes like follows:

Hi there. How are you? I feel sorry since I purchased Maple. Let see if you would agree. First of all, it is inferior to the Ti-89 in some aspects. I have tried to use Maple 11.02 to solve the problems: (sqrt(2)+1)^x+(sqrt(2)-1)^x = 3; And Maple 11.02 fail to solve, then I tried to solve numerically, it missed one solution. There must be 2 solutions for the problem above and Maple missed 1, the Ti-89 beats it hand down. The second problem I tried was: int(sqrt(sin(x))/(sqrt(sin(x))+sqrt(cos(x))), x = 0 .. (1/2)*Pi); and Maple even stuck... The Ti-89 return answer correctly within about 20 secs.

Inspired by the post Re: the physics package I decided to have a closer look at the function FeynmanDiagrams. As the Lagrangian I thought I might as well take the QED Lagrangian for a massless spinor field Q[i](X) coupled to an external electromagnetic field A[mu](X):

restart:
with(Physics):
Setup(advanced):
L_QED :=
   +Dagger(Q[i](X)) * Dgamma[4] * Dgamma[mu][i,j] * I * diff(Q[j](X),X[mu])
   +Dagger(Q[i](X)) * Dgamma[4] * Dgamma[mu][i,j] * e * A[mu](X) * (Q[j](X));

My previous blog entry was a real success. Even though my original idea about multi-part MIME has not gotten anywhere, I do now have a concise way to package a maplet with supplemental files in a single package that can be downloaded via the WWW and automatically extracted and executed. Most of the ideas were presented by acer. acer first suggested that I look at the interactive interface to the InstallerBuilder. The idea here was to embed the maplet in a worksheet saved in a help database (hdb). This did work, but was not suitable for actual use due to the overhead of the installer. In the attempt to reduce this overhead, acer then supplied some code that used march and LibraryTools. To test the product of this interaction, download the file at the URL http://www.math.sc.edu/~meade/TEST/SimpleTest.mla.

Note added: As correctly pointed out by Joe Riel in his post below, my function fromTree fails doing what it is supposed to do for expressions which contain lists and/or sets. For a version of fromTree which behaves properly (I believe) also for expressions containing lists and/or sets, see my post below. Consider the following two functions:

toTree   := x -> `if`(op(x) = x,x,[op(0,x),map(toTree,[op(x)])[]]):

I got an interesting question about integration yesterday. The question was about the integral of the rather innocuous looking function f := sqrt(1+sin(x)). The inside of the square root is always non-negative so the function is continuous (and bounded!) so it must have a continuous integral. The question I was asked, was if the following result was a bug in Maple: Int(sqrt(1+sin(x)),x) = (2*(sin(x)-1))*sqrt(1+sin(x))/cos(x) since the right-hand side is definitely not continuous at x=-Pi/2 + 2*n*Pi!

Below is a link to a file I uploaded showing an example of using assigning the value obtained from eval() to a variable and then using evalf() versus using eval() inside of evalf(). The fact that there is some difference is not surprising; it's the sheer amount of difference that amazes me--up to 100%! Do also note that the difference "settle down" over time, which is to be expected. View 413_odd_rounding_example.mw on MapleNet or