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This is the front page of the SandBox. You can try anything you like
here but keep in mind that other people are also using these pages to
learn and experiment with FriCAS and Reduce. Please be courteous to
others if you correct mistakes and try to explain what you are doing.

No Email Notices

Normally, if you edit any page on MathAction and click Save or if you add a comment to a page, a notice of the change is sent out to all subscribers on the axiom-developer email list, see the [Axiom Community]. Separate notices are also sent to those users who subscribe directly to MathAction.

Use Preview

If you click Preview instead of Save, you will get a chance to see the result of your calculations and LaTeX commands but no email notice is sent out and the result is not saved until you decide to click Save or not.

Use the SandBox

On this page or on any other page with a name beginning with SandBox such as SandBoxJohn2, SandBoxSimple, SandBoxEtc, clicking Save only sends email notices to users who subscribe directly to that specific SandBox page. Saving and adding comments does not create an email to the email list. You can safely use these pages for testing without disturbing anyone who might not care to know about your experiments.

SandBox Pages

You can also create new SandBox pages as needed just by editing this page and adding a link to the list of new page below. The link must include at least two uppercase letters and no spaces or alternatively it can be any phrase written inside [ ] brackets as long as it begins with SandBox. When you Save this page, the link to the new page will appear with a blue question mark ? beside it. Clicking on the blue question mark ? will ask you if you wish to create a new page.

[SandBox Aldor Category Theory]
based on "Prospects for Category Theory in Aldor" by Saul Youssef, 2004 http://atlas.bu.edu/~youssef/papers/math/aldor/aldor.pdf
[SandBox Aldor Foreign]
Using Aldor to call external C routines
[SandBox Aldor Generator]
Aldor defines a generator for type Vector
[SandBox Aldor Join and Meet]
Aldor has category constructor named Meet which appears to be analogous to (but opposite of) Join.
[SandBox Aldor Semantics]
exports and constants
[SandBox Aldor Sieve]
A prime number sieve in Aldor to count primes <= n.
[SandBox Aldor Testing]
Using Aldor to write Axiom library routines
[SandBox Arrays]
How fast is array access in Axiom?
[SandBox Axiom Syntax]
Syntax of if then else
[SandBox Boolean]
evaluating Boolean expressions and conditions
[SandBox Cast]
Meaning and use of pretend vs. strong typing
[SandBox Categorical Relativity]
Special relativity without the Lorentz group
[SandBox Category of Graphs]
Graph theory in Axiom
Tangle operation for literate programming implemented in Common Lisp
[SandBox Combinat]
[SandBox Content MathML]
Content vs. presentation MathML


[SandBox Direct Product]
A x B
[SandBox DistributedExpression]
expression in sum-of-products form
[SandBox Domains and Types]
What is the difference?
What does the type means for you?
Beginnings of an Emacs mode for Axiom based off of Jay's work and others
[SandBox Embeded PDF]
pdf format documents can be displayed inline
[SandBox EndPaper]
Algebra and Data Structure Hierarchy (lattice) diagrams
[SandBox Folding]
experiments with DHTML, javascript, etc.
[SandBox Functional Addition]
"adding" two functions
[SandBox Functions]
How do they work?
[SandBox Functors]
What are they? In Axiom functors are also called domain constructors.
[SandBox Gamma]
Numerical evaluation of the incomplete Gamma function
[SandBox GuessingSequence]
Guessing integer sequences
[SandBox InputForm]
How to output as inputForm
[SandBox Integration]
Examples of integration in Axiom and Reduce
[SandBox Kernel]
What is a "kernel"?

[SandBox kaveh]

[SandBox LaTeX]
LaTeX commands allowed in MathAction
[SandBox Lisp]
Using Lisp in Axiom
[SandBox Manip]
expression manipulations
[SandBox Manipulating Domains]
testing the domain of an expression
[SandBox Mapping]
A->B is a type in Axiom


[SandBox Matrix]
Examples of working with matrices in Axiom
[SandBox Maxima]
Testing the Maxima interface
[SandBox Monoid]
Rings and things
[SandBox Monoid Extend]
Martin Rubey's beautiful idea about using extend to add a category to a previously defined domain.
[SandBox Noncommutative Polynomials]
XPOLY and friends
[SandBox Numerical Integration]
Simpson method
[SandBox NNI]
NonNegative Integer without using SubDomain
[SandBox Pamphlet]
[Literate Programming] support on MathAction
Trigonometric expansion example
Computing the Pfaffian of a square matrix
[SandBox Polymake]
an interface between Axiom and PolyMake
[SandBox Polynomials]
Axiom's polynomial domains are certainly rich and complex!
[SandBox ProblemSolving]
Test page for educational purposes
[SandBox Qubic]
Solving cubic polynomials
[SandBox Reduce And MathML]
Reduce can use MathML for both input and output
[SandBox Reflection in Aldor]
a reflection framework
Slides for IARD 2006: Addition of Relative Velocites is Associative
[SandBox Risch]
Find primitive of univariate functions
[SandBox Sage]
This is a test of Sage in MathAction
[SandBox Shortcoming]
Implementation of solve
[SandBox Solve]
Solving equations
[SandBox SPAD dependent types]
SPAD: parameter-dependent types in function definitions
[SandBox Statistics]
calculating statistics in Axiom
[SandBox SubDomain]
What is a SubDomain?
[SandBox Tail Recursion]
When does Axiom replace recursion with iteration?
[SandBox Text Files]
How to access text files in Axiom
[SandBox Trace Analysed]
Tracing can affect output of 1::EXPR INT or 1::FRAC INT
[SandBox Units and Dimensions]
Scientific units and dimensions
[SandBox Spad]
Domain construction
[SandBox Speed]
Compilation speed

[Sandbox Variables Evaluation]

[SandBox Zero]

[SandBox Axiom Strengths]

Experiments with matrices and various other stuff
Experiments with GraphViz and StructuredTables
Differential Equations etc.
Here you can create your own SandBox.
Experiments with JET Bundles


Solving some nonlinear differential equations
[SandBox DoOps]
used to run Axiom without actually have to have it installed!



Maximum likelihood estimation (statistics)
Fisher's exact test for 2x2 tables (statistics)
Newton's method for numerically solving f(x)=0 (with examples of calling Axiom expressions and Spad functions from Lisp).
Test long lines
[SandBox Complementsdalgebrelineaire]
Francois Maltey
page for testing friCAS
Aldor compiler problem?
calling reduce with empty list
Category theory terminology used in SPAD
Some tests to mimic R^n
First order language over comparable types (tests for qel)
Cells and k-surfaces (preps for manifold<-charts)
Some tests for pqty buckingham pi (physical quantities)
Tensor product of three different spaces: U#V#W
Some tests for k-cells and k-surfaces (Rudin/PMMA)
Some unification tests
Jacobi diagonalization algorithm
Jacobi diagonalization algorithm (Interval Float)
Physical unit systems (FreeAbelianGroup replaced)
Book A=B
Revised simplex method
Discrete optimal transport
Jacobi diagonalisation
Roots of unity, https://en.wikipedia.org/wiki/Root_of_unity
Discrete Fourier Transform
SExpression pattern matching (unifier)
SuperSimplePatternMatcher (http://www.cs.northwestern.edu/academics/courses/325/readings/pat-match.php)
Interactive Computer Manipulation of Formal Sums (http://www.csd.uwo.ca/~watt/home/students/theses/NPatil2010-msc.pdf)
Differential forms (graded)
Test SBCL 1.1 (table matching/lookup using unification)
Demo Issues (n:PI ???)
Idea: use GradedAlgebra and TensorPower without n.
Combine monoid/module for algebra like impls.
Demo XDistributedPolynomial(B,R) (see above)
Least effort solution using XFreeAlgebra
Simplified; -GradedAlgebra; +Ring (any)



Click on the ? to create a new page. You should also edit this page to include a description and a new empty link for the next person.


Here is a simple Axiom command:

    integrate(1/(a+z^3), z=0..1,"noPole")

\begin{axiom} integrate(1/(a+z^3), z=0..1,"noPole") \end{axiom}

And here is a REDUCE command:

  load_package sfgamma;
  load_package defint;
  int(1/(a+z^3), z,0,1);

load_package sfgamma;
load_package defint;
*** gamma declared operator
int(1/(a+z^3), z,0,1);
$$\displaylines{\qdd \frac{a^{\frac{1}{ 3}}\cdot \(6\cdot \sqrt{3}\cdot \atan \(\frac{2\cdot a^{\frac{2}{ 3}}\cdot \sqrt{3} - \sqrt{3}\cdot a}{ 3\cdot a} \) + \sqrt{3}\cdot \pi +3\cdot \ln \(\frac{3\cdot a^{\frac{2}{ 3}} +3\cdot a^{\frac{1}{ 3}} +a +1}{ a +1} \) \) }{ 18\cdot a} \cr} $$

Common Mistakes

Please review the list of Common Mistakes and the list of MathAction Problems if you are have never used MathAction before. If you are learning to use Axiom and think that someone must have solved some particular problem before you, check this list of Common [Axiom Problems]?.

\begin{axiom} solve(a*x+b,x) \end{axiom}

exploring --Bill Page, Thu, 24 Apr 2008 07:00:25 -0700 reply

lexical scope --Bill Page, Sun, 11 May 2008 06:43:30 -0700 reply
Testing lexical scoping rules in SandBoxLexicalScope.

examples of overloading in SPAD --Bill Page, Fri, 16 May 2008 09:40:58 -0700 reply

Combinatorial Sum --Bill Page, Fri, 16 May 2008 13:38:20 -0700 reply
SandBoxSum (like Product)

Symbolic computations --Bill Page, Thu, 22 May 2008 12:58:29 -0700 reply

add inheritance issue --Bill Page, Sun, 25 May 2008 11:26:13 -0700 reply
For example: SandBoxLeftFreeModule

Added Preview and Cancel to comment form --page, Tue, 27 May 2008 15:08:43 -0700 reply
This is a test of the Preview and Cancel buttons: \begin{axiom} integrate(sin x, x) \end{axiom}

software archeology discovers another --Bill Page, Fri, 30 May 2008 21:01:37 -0700 reply

try iso-experiment/combinat --Bill Page, Tue, 03 Jun 2008 13:24:13 -0700 reply

Equation, Inequation, and Inequality --Bill Page, Mon, 09 Jun 2008 18:28:56 -0700 reply
SandBoxEquation? SandBoxInequation SandBoxInequality?

test aldor code --Bill Page, Wed, 18 Jun 2008 03:43:05 -0700 reply

gnuplottex --Bill Page, Tue, 24 Jun 2008 22:42:51 -0700 reply

implementing Integer from Cardinal (unsigned) numbers --Bill Page, Mon, 21 Jul 2008 06:34:23 -0700 reply

Attributes and categories --Bill Page, Fri, 25 Jul 2008 13:36:54 -0700 reply

Literal and Symbol in SPAD --Bill Page, Sun, 27 Jul 2008 02:09:57 -0700 reply

a category of partially ordered sets --Bill Page, Wed, 06 Aug 2008 17:26:50 -0700 reply

in response to an exchange of emails with Gabriel Dos Reis concerning the validity of automatic translations of x >= y into not x < y, etc.

Document function selection process in the interpreter --Bill Page, Wed, 13 Aug 2008 18:10:26 -0700 reply

Tensor Product of Polynomials --Bill Page, Sun, 24 Aug 2008 06:36:20 -0700 reply

Reflection --Bill Page, Mon, 08 Sep 2008 06:09:27 -0700 reply
\begin{spad} )abbrev package REFL Reflect Reflect(T:Type): with constructor? : Symbol -> Boolean == add constructor?(p:Symbol):Boolean == car(devaluate(T)$Lisp)$SExpression? = convert(p)$SExpression? \end{spad}

\begin{axiom} T1:=Integer T2:=Polynomial Fraction T1 T3:=Complex T2 constructor?('Polynomial)$Reflect(T1) constructor?('Polynomial)$Reflect(T2) constructor?('Polynomial)$Reflect(T3) constructor?('Complex)$Reflect(T3) \end{axiom}

Francois Maltey --Bill Page, Tue, 18 Nov 2008 19:11:35 -0800 reply

NonZeroInteger? --Bill Page, Tue, 02 Dec 2008 21:35:49 -0800 reply
SandBoxNonZeroInteger is an attempt to define the domain of Integers without 0.

Martin's "generator" mini-tutorial --Bill Page, Sat, 28 Feb 2009 09:00:14 -0800 reply

Ralf Hemmecke's example for hidden overloading --Bill Page, Mon, 23 Mar 2009 07:22:32 -0700 reply

prototype for tensor products --Bill Page, Tue, 12 May 2009 12:49:48 -0700 reply
SandBoxTensorProduct by Franz Lehner

Riemann Surface --Bill Page, Sat, 20 Jun 2009 13:15:37 -0700 reply

Differential Algebra --Bill Page, Wed, 29 Jul 2009 09:07:26 -0700 reply

Grassmann Algebra --Bill Page, Thu, 10 Sep 2009 09:05:28 -0700 reply
SandBoxGrassmannIsometry - All mappings that preserve a given metric are given in terms of the decomposition of a general multivector.

Free Product --Bill Page, Fri, 18 Sep 2009 03:20:41 -0700 reply

This domain implements the free product of monoids (or groups) It is the coproduct in the category of monoids (groups). FreeProduct(A,B) is the monoid (group) whose elements are the reduced words in A and B, under the operation of concatenation followed by reduction:

  • Remove identity elements (of either A or B)
  • Replace a1a2 by its product in A and b1b2 by its product in B

Ref: http://en.wikipedia.org/wiki/Free_product

FunctionWithCache? --Bill Page, Tue, 27 Oct 2009 14:51:49 -0700 reply
Franz Lehner provided the following example of caching the output of a function: SandBoxRemember

Hash Functions --Bill Page, Wed, 04 Nov 2009 00:11:08 -0800 reply
MortonCode (also called z-order) is a method of combining multidimensional "coordinates" into a one-dimensional coordinate or "code" that attempts to preserve locality, i.e. minimize the average Euclidean distance between coordinate locations associated with adjacent codes. Morton codes are computationally less expensive to convert to and from coordinate values than Hilbert codes.

Groebner Basis and Polynomial Ideals --Bill Page, Tue, 08 Feb 2011 14:48:48 -0800 reply
SandBoxGroebnerBasis examples from Ideals, Varieties, and Algorithms Third Edition, 2007

Frobenius Algebra --Bill Page, Fri, 11 Feb 2011 17:12:43 -0800 reply
FrobeniusAlgebraVectorSpacesAndPolynomialIdeals Classifying low dimensional Frobenius algebras

Compiling SPAD code from a string --Bill Page, Mon, 07 Mar 2011 15:59:46 -0800 reply
SandBoxSTRING2SPAD demonstrates how to call the [SPAD]? compiler from the interpreter.

Sandbox with some simple Algebra [SimplifyingAlgebraicExpressions]?

Observer algebra --Bill Page, Thu, 26 Apr 2012 14:44:52 -0700 reply

Mathematica or sympy --Bill Page, Sat, 20 Apr 2013 01:35:47 +0000 reply
FriCAS \begin{axiom} g:=1/(x(1-a(1-x))) integrate(g,x) \end{axiom}

Sympy \begin{equation} \label{eq1} \frac{- \log{\left (2 x \right )} + \log{\left (\frac{- a^{2} + 2 a x \left(a - 1\right) - a \left(a - 1\right) + 3 a - 2}{a \left(a - 1\right)} \right )}}{ a - 1} \end{equation}

Mathematica \begin{equation} \label{eq2} \frac{\log (a (x-1)+1)-\log (x)}{a-1} \end{equation}

Thomas Baruchel --Bill page, Sat, 22 Nov 2014 14:30:34 +0000 reply
\begin{axiom} )set output algebra on )set output tex off r1:=(16x^14-125x^10+150x^6+375x^2)/(256x^16+480x^12+1025x^8+750x^4 +625) r2:=integrate(r1,x=0..1); numeric r2 unparse(simplify(r2)::InputForm) r3:=integrate(r1,x); unparse(simplify(r3)::InputForm) r4:=D(r3,x); --simplify(r1-r4) --normalize(r1-r4) r5:=eval(r1-r4,x=10); numeric r5 \end{axiom}

LaTeX output formatting --Bill page, Mon, 29 Dec 2014 19:38:46 +0000 reply
\begin{axiom} )set output algebra on sin(x^b) D(%,x) D(%,x) )set output algebra off )set output tex on \end{axiom}

\begin{axiom} guessRec([1,1,0,1,- 1,2,- 1,5,- 4,29,- 13,854,- 685]?) \end{axiom}

\begin{axiom} integral(x^2, x) integrate(x^2, x) \end{axiom}

\begin{axiom} integrate(log(m*x+b),x) \end{axiom}

\begin{axiom} integrate(log(mx+b) exp(-(log(x) - mu)^2 / (2sigma^2)) / (Xsigmasqrt(2pi)), x) \end{axiom}

\begin{axiom} integrate(log(b) exp(-(log(x) - mu)^2 / (2sigma^2)) / (Xsigmasqrt(2*pi)), x) \end{axiom}

Some or all expressions may not have rendered properly, because Axiom returned the following error:
Error: export FRICAS=`aaag`; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=`aaag`/lib; LANG=en_US.UTF-8 $FRICAS/bin/FRICASsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4913059123395927175-25px.axm
/bin/sh: aaag: not found
/bin/sh: aaag: not found
/bin/sh: /bin/FRICASsys: not found

Some or all expressions may not have rendered properly, because Latex returned the following error:
This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6)
 %&-line parsing enabled.
entering extended mode
LaTeX2e <2005/12/01>
Babel <v3.8h> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov
ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon
ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i
nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp
eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia
n, swedish, ukenglish, pinyin, loaded.
Document Class: article 2005/09/16 v1.4f Standard LaTeX document class
`PSTricks' v1.15  <2006/12/22> (tvz)
`pst-plot' v1.05, 2006/11/04 (tvz,dg,hv)))
 v97 patch 2, 1999/12/12
 v1.41, 2004/05/18 <tvz>)))

Package geometry Warning: `lmargin' and `rmargin' result in NEGATIVE (-108.405p t). `width' should be shortened in length.

) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `? option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/setspace/setspace.sty Package: `setspace 6.7 <2000/12/01> ) (/usr/share/texmf-texlive/tex/generic/xypic/xy.sty (/usr/share/texmf-texlive/tex/generic/xypic/xy.tex Bootstrap'ing: catcodes, docmode, (/usr/share/texmf-texlive/tex/generic/xypic/xyrecat.tex) (/usr/share/texmf-texlive/tex/generic/xypic/xyidioms.tex)

Xy-pic version 3.7 <1999/02/16> Copyright (c) 1991-1998 by Kristoffer H. Rose <krisrose@ens-lyon.fr> Xy-pic is free software: see the User's Guide for details.

Loading kernel: messages; fonts; allocations: state, direction, utility macros; pictures: \xy, positions, objects, decorations; kernel objects: directionals, circles, text; options; algorithms: directions, edges, connections; Xy-pic loaded)) (/usr/share/texmf-texlive/tex/generic/xypic/xyall.tex Xy-pic option: All features v.3.3 (/usr/share/texmf-texlive/tex/generic/xypic/xycurve.tex Xy-pic option: Curve and Spline extension v.3.7 curve, circles, loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyframe.tex Xy-pic option: Frame and Bracket extension v.3.7 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xycmtip.tex Xy-pic option: Computer Modern tip extension v.3.3 (/usr/share/texmf-texlive/tex/generic/xypic/xytips.tex Xy-pic option: More Tips extension v.3.3 loaded) loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyline.tex Xy-pic option: Line styles extension v.3.6 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyrotate.tex Xy-pic option: Rotate and Scale extension v.3.3 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xycolor.tex Xy-pic option: Colour extension v.3.3 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xymatrix.tex Xy-pic option: Matrix feature v.3.4 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyarrow.tex Xy-pic option: Arrow and Path feature v.3.5 path, \ar, loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xygraph.tex Xy-pic option: Graph feature v.3.7 loaded) loaded) (/usr/share/texmf-texlive/tex/latex/tools/verbatim.sty) (/usr/share/texmf/tex/latex/graphviz/graphviz.sty (/usr/share/texmf-texlive/tex/latex/psfrag/psfrag.sty)) (/usr/share/texmf/tex/latex/sagetex.sty Writing sage input file 2401772337697193154-16.0px.sage ) (/usr/share/texmf-texlive/tex/latex/gnuplottex/gnuplottex.sty (/usr/share/texmf-texlive/tex/latex/base/latexsym.sty) (/usr/share/texmf-texlive/tex/latex/moreverb/moreverb.sty) (/usr/share/texmf-texlive/tex/latex/base/ifthen.sty)

Package gnuplottex Warning: Shell escape not enabled. (gnuplottex) You'll need to convert the graphs yourself..

) (./2401772337697193154-16.0px.aux) (/usr/share/texmf-texlive/tex/latex/ucs/ucsencs.def) (/usr/share/texmf-texlive/tex/latex/base/t1cmtt.fd)

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 123.

(/usr/share/texmf-texlive/tex/latex/jknapltx/ursfs.fd) (/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd) (/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd) (/usr/share/texmf-texlive/tex/latex/base/ulasy.fd)

Package amsmath Warning: Foreign command \over; (amsmath) \frac or \genfrac should be used instead (amsmath) on input line 151.


LaTeX Warning: Characters dropped after `\end{axiom}' on input line 155.

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 158.

Missing $ inserted. <inserted text> $ l.165 \end{spad} \newpage [2]

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 174.

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 178.

[3] [4]

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 205.

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 213.

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 216.

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 220.

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 223.

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 226.

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 229.

[5] (./2401772337697193154-16.0px.aux) ) (see the transcript file for additional information) Output written on 2401772337697193154-16.0px.dvi (5 pages, 4548 bytes). Transcript written on 2401772337697193154-16.0px.log.

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