\documentclass{book}
%\newcommand{\VolumeName}{Volume 2: Axiom Users Guide}
%\input{bookheader.tex}
\pagenumbering{arabic}
\mainmatter
\setcounter{chapter}{0} % Chapter 1

\usepackage{makeidx}
\makeindex
\begin{document}
\begin{verbatim}
\start
Date: Fri, 2 Oct 2015 08:06:59 +0200 (CEST)
From: Waldek Hebisch <hebisch@math.uni.wroc.pl>
To: daly@axiom-developer.org
Subject: Re: [Axiom-developer] Compiling Axiom on Ubuntu 14.04, 64 bit

Tim Daly wrote:
> Alasdair McAndrew wrote:
> >Finally, I believe that Axiom is the only open-source software which
> >includes a complete implementation of the Risch decision algorithm for
> >symbolic integration; done by the late Manuel Bronstein initially in the
> >1970's and 1980's.  Is this correct?  I suppose that the major commercial
> >systems support as complete integration routines as possible, but Axiom has
> >the edge on other (free) systems as far as I know.
> 
> As far as I know Axiom has the "most complete" implementation.
> There are still cases which are not implemented but Manuel did more
> that anyone else.

Actually, while "most complete" when written Bronstain's implementation
contained substantial gaps.  FriCAS contains significant enhancement
of Bronstain's code.  AFAICS FriCAS is the only system which can
resonably claim completeness in purely transcendental case.
For the old code is is relatively to build examples that either
signal internal errors or return unevaluated.
Completing this case required about 1500 lines, while in Bronstain's
version about 2500 lines handles transcendental case.  So about 30%
of code was missing.

In fact in FriCAS more than 25% of integration code is new.  IIUC
only tiny part of this (a few bug fixes) is included in Axiom.
So FriCAS can claim to have "most complete" implementation, but
Axiom no longer can.

Concerning commercial systems, Maple claim to implement
Trager algoritm and transcendental part of Risch algorithm,
which is about what Bronstain's claimed to implement.
However, at least with default Maple settings old Bronstain's
examples still return unevaluated.  IIRC when testing
FriCAS implementation of transcendental case I found
some other cases which Maple returns unevaluated.
Mathematica makes far reaching but imprecise claim
("almost any function"), but experimental results are
similar to Maple.

AFAICS commercial systems take mostly ad hoc approach to
integration, they probably contain a lot of special case
code (pattern, tables or routines handling some narrow
class of functions).  In simple cases this produces
impressive result.  However, it is also fairly incomplete.
FriCAS not contains extension of Risch algorithm to
special functions (integrals needing Ei, erf, Gamma, ...).
When comparing FriCAS with Ma-s I noticed that once
examples got complicated enough, then Ma-s could no
longer do them.

-- 
                              Waldek Hebisch

\start
Date: Fri, 2 Oct 2015 18:47:42 +1000
From: Alasdair McAndrew <amca01@gmail.com>
To: Waldek Hebisch <hebisch@math.uni.wroc.pl>
Subject: Re: [Axiom-developer] Compiling Axiom on Ubuntu 14.04, 64 bit

Many thanks for your responses, and for your comments.

Is there a list somewhere of integrals which can be solved in closed form
by Axiom/FriCAS, but cannot be solved by the most recent versions of
Maple/Mathematica?  I would be very interested in seeing such a list - but
I don't myself have access at the moment to either Maple or Mathematica.

Many thanks,
Alasdair

On Fri, Oct 2, 2015 at 4:06 PM, Waldek Hebisch <hebisch@math.uni.wroc.pl>
wrote:

> Tim Daly wrote:
> > Alasdair McAndrew wrote:
> > >Finally, I believe that Axiom is the only open-source software which
> > >includes a complete implementation of the Risch decision algorithm for
> > >symbolic integration; done by the late Manuel Bronstein initially in the
> > >1970's and 1980's.  Is this correct?  I suppose that the major
> commercial
> > >systems support as complete integration routines as possible, but Axiom
> has
> > >the edge on other (free) systems as far as I know.
> >
> > As far as I know Axiom has the "most complete" implementation.
> > There are still cases which are not implemented but Manuel did more
> > that anyone else.
>
> Actually, while "most complete" when written Bronstain's implementation
> contained substantial gaps.  FriCAS contains significant enhancement
> of Bronstain's code.  AFAICS FriCAS is the only system which can
> resonably claim completeness in purely transcendental case.
> For the old code is is relatively to build examples that either
> signal internal errors or return unevaluated.
> Completing this case required about 1500 lines, while in Bronstain's
> version about 2500 lines handles transcendental case.  So about 30%
> of code was missing.
>
> In fact in FriCAS more than 25% of integration code is new.  IIUC
> only tiny part of this (a few bug fixes) is included in Axiom.
> So FriCAS can claim to have "most complete" implementation, but
> Axiom no longer can.
>
> Concerning commercial systems, Maple claim to implement
> Trager algoritm and transcendental part of Risch algorithm,
> which is about what Bronstain's claimed to implement.
> However, at least with default Maple settings old Bronstain's
> examples still return unevaluated.  IIRC when testing
> FriCAS implementation of transcendental case I found
> some other cases which Maple returns unevaluated.
> Mathematica makes far reaching but imprecise claim
> ("almost any function"), but experimental results are
> similar to Maple.
>
> AFAICS commercial systems take mostly ad hoc approach to
> integration, they probably contain a lot of special case
> code (pattern, tables or routines handling some narrow
> class of functions).  In simple cases this produces
> impressive result.  However, it is also fairly incomplete.
> FriCAS not contains extension of Risch algorithm to
> special functions (integrals needing Ei, erf, Gamma, ...).
> When comparing FriCAS with Ma-s I noticed that once
> examples got complicated enough, then Ma-s could no
> longer do them.
>
> --
>                               Waldek Hebisch
>



-- 
[image: http://www.facebook.com/alasdair.mcandrew]
<http://www.facebook.com/alasdair.mcandrew> [image:
https://plus.google.com/+AlasdairMcAndrew/posts]
<https://plus.google.com/+AlasdairMcAndrew/posts> [image:
https://www.linkedin.com/pub/alasdair-mcandrew/a/178/108]
<https://www.linkedin.com/pub/alasdair-mcandrew/a/178/108> [image:
https://twitter.com/amca01] <https://twitter.com/amca01> [image:
http://numbersandshapes.net] <http://numbersandshapes.net>

\start
Date: Fri, 2 Oct 2015 11:34:25 -0500
From: daly@axiom-developer.org
To: Alasdair McAndrew <amca01@gmail.com>
Subject: Re: [Axiom-developer] Compiling Axiom on Ubuntu 14.04, 64 bit

>Is there a list somewhere of integrals which can be solved in closed form
>by Axiom/FriCAS, but cannot be solved by the most recent versions of
>Maple/Mathematica?  I would be very interested in seeing such a list - but
>I don't myself have access at the moment to either Maple or Mathematica.

See the computer algebra test suite page:
http://axiom-developer.org/axiom-website/CATS/index.html

It contains Axiom's answers for Schaums integrals, the Charlwood integrals,
and many of the Rich integrals, about 18000 integration tests so far. There
are also tests of ordinary differential equations, the Kamke test suite. If
you know of any other test suites I would be interested.

Albert Rich (see the sci.math.symbolic mailing list) has tested his set
on all of the systems. MMA and Maple do quite well on his test suite,
well beyond what Axiom can do, for non-Risch integrals.

I have been working with Albert to implement his machinery although
none of that work is yet public. The plan is to be able to do all of
his integrals (about 50000).

Tim

\start
Date: Fri, 2 Oct 2015 18:51:11 +0200 (CEST)
From: Waldek Hebisch <hebisch@math.uni.wroc.pl>
To: amca01@gmail.com (Alasdair McAndrew)
Subject: Re: [Axiom-developer] Hard integrals

Alasdair McAndrew wrote:
> 
> Is there a list somewhere of integrals which can be solved in closed form
> by Axiom/FriCAS, but cannot be solved by the most recent versions of
> Maple/Mathematica?  I would be very interested in seeing such a list - but
> I don't myself have access at the moment to either Maple or Mathematica.

Personaly I have access to Maple 15 and Mathematica 8, which are
not the newest versions.  At

http://12000.org/my_notes/CAS_integration_tests/index.htm

Nasser M. Abbasi posted results of running Maple 2015.1 and
Mathematica 10.1 trough Rubi testsuite.  Below are some
examples from this testsuite, which Nasser reports as
unevaluated by Mathematica and Maple.  Note: Rubi testsuite
strongly favours pattern matchers and does not contain
many difficulties which complete algorithm is supposed to
handle.  Still, for some classes of integrals which FriCAS
can do in systematic way you can find there examples
which are beyond Mathematica and Maple.  BTW:
I can provide several examples which are beyond Rubi,
Maple 15 and Mathematica 8, but can not check them
with newer versions.

integrate(f^(c*(b*x+a)^3)*x^2,x)

   (1)
           3   3       2   2     2         3          +------------+2
         (b c x  + 3a b c x  + 3a b c x + a c)log(f) 3|   3
       %e                                            \|- b c log(f)
     + 
       -
             2                1     3   3       2   2     2         3
            a b c log(f)Gamma(-,(- b c x  - 3a b c x  - 3a b c x - a c)log(f))
                              3
         *
             +------------+
            3|   3
            \|- b c log(f)
     + 
           2              2     3   3       2   2     2         3
       2a b c log(f)Gamma(-,(- b c x  - 3a b c x  - 3a b c x - a c)log(f))
                          3
  /
                 +------------+2
       3        3|   3
     3b c log(f)\|- b c log(f)
                                         Type: Union(Expression(Integer),...)


integrate(exp(b^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3)*x^4,x)

   (2)
                                3 3       2 2     2       3  +----+2
          2 2              2   b x  + 3a b x  + 3a b x + a  3|   3
       (3b x  - 6a b x + 9a )%e                             \|- b
     + 
                                                             +----+
            4              1    3 3       2 2     2       3 3|   3
       (- 3a  - 4a)b Gamma(-,- b x  - 3a b x  - 3a b x - a )\|- b
                           3
     + 
           3      2      2    3 3       2 2     2       3
       (12a  + 2)b Gamma(-,- b x  - 3a b x  - 3a b x - a )
                         3
  /
          +----+2
       5 3|   3
     9b  \|- b
                                         Type: Union(Expression(Integer),...)


integrate(exp(e*(d*x+c)^3)*(b*x+a)^3,x)                

   (3)
                                  3   3       2   2     2         3   +-----+2
          3          2      3    d e x  + 3c d e x  + 3c d e x + c e 3|   3
       (3b d x + 9a b d - 6b c)%e                                    \|- d e
     + 
               3 4     2     3       2 2 2     3 3       3
         ((- 3a d  + 9a b c d  - 9a b c d  + 3b c d)e + b d)
      *
                                                        +-----+
               1    3   3       2   2     2         3  3|   3
         Gamma(-,- d e x  - 3c d e x  - 3c d e x - c e)\|- d e
               3
     + 
              2   4        2   3     3 2 2
         (- 9a b d  + 18a b c d  - 9b c d )e
      *
               2    3   3       2   2     2         3
         Gamma(-,- d e x  - 3c d e x  - 3c d e x - c e)
               3
  /
           +-----+2
       4  3|   3
     9d e \|- d e
                                         Type: Union(Expression(Integer),...)
-- 
                              Waldek Hebisch

\start
Date: Mon, 5 Oct 2015 21:31:07 -0500
From: daly@axiom-developer.org
To: <amca01@gmail.com>
Subject: [Axiom-developer] Denesting radicals?

Denesting radicals is a deep problem. There are subtle issues
when you are using symbols and solving equations. If I remember
correctly, one of the issues is that you can't decide if something
is square-free if it contains certain kinds of radicals (don't 
depend on my memory though). 

Simplification of radicals usually involves algebraic field extensions
but (again, if I remember correctly) the issue is independence. Doing
a field extension with sqrt(2) and sqrt(6) causes problems.

Plus the question arises whether you're working over real or complex.

We had these discussions in the last century (around 1980) and I don't
remember all the details except to say that it is MUCH harder and
more subtle than it looks.

The person I'd ask is either Barry Trager or James Davenport.

Tim

\start
Date: Thu, 1 Oct 2015 22:54:18 -0500
From: daly@axiom-developer.org
To: Alasdair McAndrew <amca01@gmail.com>
Subject: Re: [Axiom-developer] Compiling Axiom on Ubuntu 14.04, 64 bit

>The instructions available here:
>
>http://www.axiom-developer.org/axiom-website/download.html
>
>include the initial step of setting randomize_va_space to 0, with:
>
>echo 0 >/proc/sys/kernel/randomize_va_space

No, this is no longer necessary. I will remove it.

>Also, are there any GUIs for axiom other that Texmacs (which I've found in
>the past to be unstable and prone to freezing)?

Axiom has a web browser front end that ''works'' in that you can
type an input and it will open a <div> to show the output. There is
still a great deal of work to be done to port the pages from the
hyperdoc browser. The pages are created during the build in
mnt/ubuntu/doc/hypertex/*.xhtml. The last thing done on that effort
was to try to implement web-sockets so the interface was reactive.

All of the machinery has been demonstrated but it is still only
demo-ware at this stage.

One of the project goals is to re-implement (and improve) the
hyperdoc machinery. Also in-plan is to re-implement the graphics
using the browser <canvas> tag. There also needs to be a tight
connection between the browser and the pdfs so you can "drill down"
on any piece of algebra by viewing the source code and explanation.
As usual, pieces exist.

>Finally, I believe that Axiom is the only open-source software which
>includes a complete implementation of the Risch decision algorithm for
>symbolic integration; done by the late Manuel Bronstein initially in the
>1970's and 1980's.  Is this correct?  I suppose that the major commercial
>systems support as complete integration routines as possible, but Axiom has
>the edge on other (free) systems as far as I know.

As far as I know Axiom has the "most complete" implementation.
There are still cases which are not implemented but Manuel did more
that anyone else.

Tim

\start
Date: Mon, 5 Oct 2015 21:07:57 -0500
From: daly@axiom-developer.org
To: <amca01@gmail.com>
Subject: Re: [Axiom-developer] Fonts for HyperDoc?

Fonts like Rom14, Erg14, etc are available at
http://www.mit.edu/afs.new/athena/system/rs_aix32/os/usr/lpp/X11fnt/X11fnt.iso88595.fnt.inventory

Alternatively hyper.h could be patched at the lines
#define RmFontDefault "Rom14" to be your favorite font.

Fonts for the Axiom browser are cached in the zips/axiomfonts.tgz file
but I haven't cached the X11 fonts. I'll look at applying the patch above
by default.

Tim

\start
Date: Mon, 5 Oct 2015 21:38:54 -0500
From: daly@axiom-developer.org
To: <amca01@gmail.com>
Subject: Re: [Axiom-developer] Fonts for HyperDoc?

Actually, I just checked the code in the Hyperdoc volume
(books/bookvol7.pamphlet).

If LINUXplatform is defined (which it is in the top level Makefile)
then

#define RmFontDefault "-adobe-courier-medium-r-normal--18-*-*-*-m-*-iso8859-1"

is used. If your AXIOM variable reads

export AXIOM=~/axiom/mnt/ubuntu

(that is, it includes the string "ubuntu" at the end then you should see 

PLF:=LINUXplatform

in the top level Makefile.

So something is failing. Do you have the adobe fonts installed?

Tim

\start
Date: Thu, 8 Oct 2015 00:07:36 -0500
From: daly@axiom-developer.org
To: "Alasdair McAndrew" <amca01@gmail.com>
Subject: Re: [Axiom-developer] Numerical Libraries

Axiom has a numerics volume (books/bookvol10.5.pamphlet) which
has code for BLAS (levels 1,2, and 3) and LAPACK. Not all of the
code runs yet. The plan is for Axiom to have full support for
BLAS and LAPACK. Try

  )show BLAS1

There are additional piles of numeric code sitting on the sidelines 
in the "to be added" queue.

Time, time, time ...

Tim

\start
Date: Thu, 8 Oct 2015 18:12:32 -0500
From: daly@axiom-developer.org
To: Alasdair McAndrew <amca01@gmail.com>
Subject: Re: [Axiom-developer] compile errors

git clone git://github.com/daly/axiom
cd axiom
export AXIOM=`pwd`/mnt/ubuntu
export PATH=$AXIOM/bin:$PATH
make

\start
Date: Thu, 8 Oct 2015 00:09:10 -0500
From: daly@axiom-developer.org
To: "Alasdair McAndrew" <amca01@gmail.com>
Subject: Re: [Axiom-developer] Starting Hyperdoc from TeXmacs

I suspect the problem is the command line TeXmacs uses to start Axiom.
I don't have it installed. Can you find the command line? Does it 
include the -nox flag? (i.e no X11)

Tim

\start
Date: Thu, 8 Oct 2015 16:15:15 +1100
From: Alasdair McAndrew <amca01@gmail.com>
To: Tim Daly <daly@axiom-developer.org>, axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] Numerical Libraries

I noticed that Axiom has some interfaces to the BLAS and LAPACK libraries:
what I was wondering if there's some sort of general mechanism by which a
bone-headed user like me can link external libraries to Axiom.  I think
that would be a great boon.

many thanks,
Alasdair

On Thu, Oct 8, 2015 at 4:07 PM, <daly@axiom-developer.org> wrote:

> Axiom has a numerics volume (books/bookvol10.5.pamphlet) which
> has code for BLAS (levels 1,2, and 3) and LAPACK. Not all of the
> code runs yet. The plan is for Axiom to have full support for
> BLAS and LAPACK. Try
>
>   )show BLAS1
>
> There are additional piles of numeric code sitting on the sidelines
> in the "to be added" queue.
>
> Time, time, time ...
>
> Tim
>



-- 
[image: http://www.facebook.com/alasdair.mcandrew]
<http://www.facebook.com/alasdair.mcandrew> [image:
https://plus.google.com/+AlasdairMcAndrew/posts]
<https://plus.google.com/+AlasdairMcAndrew/posts> [image:
https://www.linkedin.com/pub/alasdair-mcandrew/a/178/108]
<https://www.linkedin.com/pub/alasdair-mcandrew/a/178/108> [image:
https://twitter.com/amca01] <https://twitter.com/amca01> [image:
http://numbersandshapes.net] <http://numbersandshapes.net>


\start
Date: Fri, 9 Oct 2015 01:36:54 -0500
From: daly@axiom-developer.org
To: Alasdair McAndrew <amca01@gmail.com>
Subject: Re: [Axiom-developer] compile errors

I built a new Ubuntu 10.04 machine from scratch.
I installed the packages listed on the downloads web page on
axiom-developer
I did the git clone and subsequent instructions above.
My compile worked.

Perhaps your GCC is downlevel?

Tim

>Sorry - just tried that on my work computer (Ubuntu 14.04), and same errors.
>
>-Alasdair
>
>On Fri, Oct 9, 2015 at 10:18 AM, Alasdair McAndrew <amca01@gmail.com> wrote:
>
>> Thanks!  Will give it my best shot when I get home.
>>
>> cheers,
>> Alasdair
>>
>> On Fri, Oct 9, 2015 at 10:12 AM, <daly@axiom-developer.org> wrote:
>>
>>> git clone git://github.com/daly/axiom
>>> cd axiom
>>> export AXIOM=`pwd`/mnt/ubuntu
>>> export PATH=$AXIOM/bin:$PATH
>>> make

\start
Date: Fri, 9 Oct 2015 21:58:31 +0000
From: Lawrence Bottorff <borgauf@gmail.com>
To: Tim Daly <daly@axiom-developer.org>, axiom-developer@nongnu.org
Subject: Re: [Axiom-developer] compile errors

After crashing my computer a few times (Ubuntu 15.04), I finally got this:

Unrecoverable error: invocation history stack overflow.
Aborted (core dumped)
Makefile:175: recipe for target
'/home/hercynian/gitstuff/axiom/obj/ubuntu/bin/depsys' failed
make[3]: *** [/home/hercynian/gitstuff/axiom/obj/ubuntu/bin/depsys] Error
134
rm /home/hercynian/gitstuff/axiom/int/interp/nocompil.lisp
/home/hercynian/gitstuff/axiom/int/interp/sys-pkg.lisp
/home/hercynian/gitstuff/axiom/int/interp/util.lisp
/home/hercynian/gitstuff/axiom/int/interp/parsing.lisp
make[3]: Leaving directory '/home/hercynian/gitstuff/axiom/src/interp'
Makefile:79: recipe for target 'interpdir' failed
make[2]: *** [interpdir] Error 2
make[2]: Leaving directory '/home/hercynian/gitstuff/axiom/src'
Makefile.ubuntu:148: recipe for target 'srcdir' failed
make[1]: *** [srcdir] Error 2
make[1]: Leaving directory '/home/hercynian/gitstuff/axiom'
Makefile:134: recipe for target 'all' failed
make: *** [all] Error 2

Please advise.

LB

On Fri, Oct 9, 2015 at 6:36 AM, <daly@axiom-developer.org> wrote:

> I built a new Ubuntu 10.04 machine from scratch.
> I installed the packages listed on the downloads web page on
> axiom-developer
> I did the git clone and subsequent instructions above.
> My compile worked.
>
> Perhaps your GCC is downlevel?
>
> Tim
>
> >Sorry - just tried that on my work computer (Ubuntu 14.04), and same
> errors.
> >
> >-Alasdair
> >
> >On Fri, Oct 9, 2015 at 10:18 AM, Alasdair McAndrew <amca01@gmail.com>
> wrote:
> >
> >> Thanks!  Will give it my best shot when I get home.
> >>
> >> cheers,
> >> Alasdair
> >>
> >> On Fri, Oct 9, 2015 at 10:12 AM, <daly@axiom-developer.org> wrote:
> >>
> >>> git clone git://github.com/daly/axiom
> >>> cd axiom
> >>> export AXIOM=`pwd`/mnt/ubuntu
> >>> export PATH=$AXIOM/bin:$PATH
> >>> make


\start
Date: Sat, 17 Oct 2015 03:44:49 -0500
From: daly@axiom-developer.org
To: Alasdair McAndrew <amca01@gmail.com>
Subject: Re: [Axiom-developer] A question about differential equations

It is rarely the case that algorithms used for hand solutions match
the algorithms used by Axiom (or by computer algebra systems in general).

The Computer Algebra Test Suite (CATS) at:
http://axiom-developer.org/axiom-website/CATS/index.html
has an ODE section (Kamke) which shows Axiom's ODE abilities. 

These are presented in three different forms, 
 * as text files which can be used as input, 
    http://axiom-developer.org/axiom-website/CATS/kamke0.txt
 * as source files which are latex, 
    http://axiom-developer.org/axiom-website/CATS/kamke0.input.pamphlet
 * and as PDFs which are the output of the latex.
    http://axiom-developer.org/axiom-website/CATS/kamke0.input.pdf

There are occasions when the Axiom output did not simplify to the
result given in the textbook. At those times I used RewriteRule to
apply simplifications. 

Look at, for example:

http://axiom-developer.org/axiom-website/CATS/kamke0.input.pamphlet
for the expression ODE6 (--S 24) which gives an example of using RewriteRule
to manipulate the expression.
   ode6 := D(y(x),x) + y(x)*cos(x) - sin(2*x)/2
   ode6a:=solve(ode6,y,x)
   yx:=ode6a.particular
   ode6expr:=D(yx,x) + yx*cos(x) - sin(2*x)/2
   sin2rule := rule 2*cos(x)*sin(x) == sin(2*x)
   sin2rule ode6expr

http://axiom-developer.org/axiom-website/CATS/kamke1.input.pamphlet
Look at ODE94 (in kamke1) where I use
   ode94 := x*D(y(x),x) +a*y(x) + b*x**n
   ode94a:=solve(ode94,y,x)
   yx:=ode94a.particular
   ode94expr := x*D(yx,x) +a*yx + b*x**n
   exprule := rule x^n == %e^(n*log(x))
   exprule ode94expr

I'm not sure the exact form of the result you're trying to achieve
but it is likely that you can use RewriteRule to get there.

Tim


>In putting Axiom through its paces just recently (yes: Axiom, not a fork!),
>I experimented with the ODE
>
>y''+6y'+5y = 10x^2+4x+4exp(-x)
>
>Now standard techniques (such as I teach my students), produce a solution
>of the form
>
>y = A*exp(-5x)+B*exp(-x)+2*x^2-4*x+4+x*exp(-x).
>
>This is Axiom:
>
>--(View this section in a fixed width font if it isn't shown as such)
>
>(1) -> y:=operator 'y
>(2) -> deq:=D(y(x),x,2)+6*D(y(x),x)+5*y(x)=10*x^2+4*x+4*exp(-x)
>(3) -> sol:=solve(deq,y,x)
>   (3)
>   [
>       particular =
>             - x 6       2               - x 5     - 5x  - x     2  - 5x
>       4x (%e   )  + (10x  - 16x + 16)(%e   )  - %e    %e    - 2x %e
>       -----------------------------------------------------------------
>                                       - x 5
>                                   4(%e   )
>     ,
>              - x   - 5x
>    basis= [%e   ,%e    ]]
>Type: Union(Record(particular: Expression(Integer),basis:
>List(Expression(Integer)))
>
>-- 
>of which the particular solution is a bit of a jumble.   It doesn't seem to
>be particularly simplifiable:
>
>(4) -> simplifyExp(sol.particular)
>(4) ->
>           2              - 5x             - 6x
>        (8x  - 16x + 16)%e     + (4x - 1)%e
>   (4)  ---------------------------------------
>                           - 5x
>                        4%e
>
>-- 
>Is there some way I can coerce Axiom into giving a particular solution to
>such a straightforward ODE in a more simplified form?
>
>

\start
Date: Sat, 17 Oct 2015 17:04:18 +0200 (CEST)
From: Waldek Hebisch <hebisch@math.uni.wroc.pl>
To: daly@axiom-developer.org
Subject: Re: [Axiom-developer] A question about differential equations

> >In putting Axiom through its paces just recently (yes: Axiom, not a fork!),
> >I experimented with the ODE
> >
> >y''+6y'+5y = 10x^2+4x+4exp(-x)
> >
> >Now standard techniques (such as I teach my students), produce a solution
> >of the form
> >
> >y = A*exp(-5x)+B*exp(-x)+2*x^2-4*x+4+x*exp(-x).
> >
> >This is Axiom:
> >
> >--(View this section in a fixed width font if it isn't shown as such)
> >
> >(1) -> y:=operator 'y
> >(2) -> deq:=D(y(x),x,2)+6*D(y(x),x)+5*y(x)=10*x^2+4*x+4*exp(-x)
> >(3) -> sol:=solve(deq,y,x)
> >   (3)
> >   [
> >       particular =
> >             - x 6       2               - x 5     - 5x  - x     2  - 5x
> >       4x (%e   )  + (10x  - 16x + 16)(%e   )  - %e    %e    - 2x %e
> >       -----------------------------------------------------------------
> >                                       - x 5
> >                                   4(%e   )
> >     ,
> >              - x   - 5x
> >    basis= [%e   ,%e    ]]
> >Type: Union(Record(particular: Expression(Integer),basis:
> >List(Expression(Integer)))
> >
> >-- 
> >of which the particular solution is a bit of a jumble.   It doesn't seem to
> >be particularly simplifiable

try

eval(sol.particular, exp(-5*x)= exp(-x)^5)

-- 
                              Waldek Hebisch

\start
Date: Sun, 18 Oct 2015 03:29:59 +1100
From: Alasdair McAndrew <amca01@gmail.com>
To: Waldek Hebisch <hebisch@math.uni.wroc.pl>
Subject: Re: [Axiom-developer] A question about differential equations

Thank you!  In fact

normalize(sol.particular)

gives the same result.  But there's still an exp(-x) term in the particular
solution, which shouldn't be there, as exp(-x) is a basis solution.  So I
need some way of removing basis functions from the particular solution -
which possibly should be automatic.

On Sun, Oct 18, 2015 at 2:04 AM, Waldek Hebisch <hebisch@math.uni.wroc.pl>
wrote:

> > >In putting Axiom through its paces just recently (yes: Axiom, not a
> fork!),
> > >I experimented with the ODE
> > >
> > >y''+6y'+5y = 10x^2+4x+4exp(-x)
> > >
> > >Now standard techniques (such as I teach my students), produce a
> solution
> > >of the form
> > >
> > >y = A*exp(-5x)+B*exp(-x)+2*x^2-4*x+4+x*exp(-x).
> > >
> > >This is Axiom:
> > >
> > >--(View this section in a fixed width font if it isn't shown as such)
> > >
> > >(1) -> y:=operator 'y
> > >(2) -> deq:=D(y(x),x,2)+6*D(y(x),x)+5*y(x)=10*x^2+4*x+4*exp(-x)
> > >(3) -> sol:=solve(deq,y,x)
> > >   (3)
> > >   [
> > >       particular =
> > >             - x 6       2               - x 5     - 5x  - x     2  - 5x
> > >       4x (%e   )  + (10x  - 16x + 16)(%e   )  - %e    %e    - 2x %e
> > >       -----------------------------------------------------------------
> > >                                       - x 5
> > >                                   4(%e   )
> > >     ,
> > >              - x   - 5x
> > >    basis= [%e   ,%e    ]]
> > >Type: Union(Record(particular: Expression(Integer),basis:
> > >List(Expression(Integer)))
> > >
> > >--
> > >of which the particular solution is a bit of a jumble.   It doesn't
> seem to
> > >be particularly simplifiable
>
> try
>
> eval(sol.particular, exp(-5*x)= exp(-x)^5)
>
> --
>                               Waldek Hebisch
>



-- 
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<http://www.facebook.com/alasdair.mcandrew> [image:
https://plus.google.com/+AlasdairMcAndrew/posts]
<https://plus.google.com/+AlasdairMcAndrew/posts> [image:
https://www.linkedin.com/pub/alasdair-mcandrew/a/178/108]
<https://www.linkedin.com/pub/alasdair-mcandrew/a/178/108> [image:
https://twitter.com/amca01] <https://twitter.com/amca01> [image:
http://numbersandshapes.net] <http://numbersandshapes.net>

\start
Date: Tue, 20 Oct 2015 14:56:41 -0400
From: Camm Maguire <camm@maguirefamily.org>
To: daly@axiom-developer.org
Subject: [Axiom-developer] bug in server code

Greetings!  

make_server_name in sockio-c.pamphlet makes a name that can be larger
than the sa_data field of the BSD sockaddr structure and overflow the
buffer, which is only 14 bytes long.  You might want to consider
sockaddr_un.

This was uncovered on ppc64 using FORTIFY_SOURCE=2.  A crude
workaround is included below, but it can surely be made better.

Take care,
=============================================================================
int 
make_server_name(char *name,char * base)
{
  char *num;
  struct sockaddr addr;
  if (spad_server_number != -1) {
    snprintf(name, sizeof(addr.sa_data),"%s%d", base, spad_server_number);
    return 0;
  }
  num = getenv("SPADNUM");
  if (num == NULL) {
/*    fprintf(stderr,
      "\n(AXIOM Sockets) The AXIOM server number is undefined.\n");
*/
    return -1;
  }
  snprintf(name, sizeof(addr.sa_data),"%s%s", base, num);
  return 0;
}
=============================================================================
-- 
Camm Maguire			     		    camm@maguirefamily.org


\end{verbatim}
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\addcontentsline{toc}{chapter}{Bibliography}
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\bibliography{axiom}
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\cleardoublepage
%\phantomsection
\addcontentsline{toc}{chapter}{Index}
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\end{document}
