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# Edit detail for SandBox revision 104 of 150

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changed:
-learn and experiment with Axiom and Reduce. Please be curteous to
learn and experiment with Axiom and Reduce. Please be courteous to

[SandBoxCategoryTerms] -- Category theory terminology used in SPAD

changed:
-SandBoxFrobeniusAlgebra Classifying low dimensional Frobenius algebras
FrobeniusAlgebraVectorSpacesAndPolynomialIdeals Classifying low dimensional Frobenius algebras

Sandbox with some simple Algebra
[SimplifyingAlgebraicExpressions]

From BillPage Thu Apr 26 14:44:52 -0700 2012
From: Bill Page
Date: Thu, 26 Apr 2012 14:44:52 -0700
Subject: Observer algebra
Message-ID: <20120426144452-0700@axiom-wiki2.newsynthesis.org>

SandBoxObserverAsIdempotent


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 Axiom 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
New SandBox Pages
You can also create new SandBox pages as needed just by
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
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
[SandBoxCL-WEB]
Tangle operation for literate programming implemented in Common Lisp
[SandBox Combinat]
A{ld,xi}o{r,m}Combinat
[SandBox Content MathML]
Content vs. presentation MathML

SandBoxCS224

[SandBox Direct Product]
A x B
[SandBox DistributedExpression]
expression in sum-of-products form
[SandBox Domains and Types]
What is the difference?
[SandboxTypeDefinitions]
What does the type means for you?
[AxiomEmacsMode]
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 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 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

[MathMLFormat]

[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
[SandBoxPartialFraction]
Trigonometric expansion example
SandBoxPfaffian
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
[SandBoxRelativeVelocity]
Slides for IARD 2006: Addition of
Relative Velocites is Associative
[SandBox Sage]
This is a test of Sage in MathAction
[SandBox Shortcoming]
Implementation of solve
[SandBox Solve]
Solving equations
[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
Domain construction
[SandBox Speed]
Compilation speed

[SandBox Zero]
[SandBox Axiom Strengths]

SandBoxJohn2
Experiments with matrices and various other stuff
SandBox2
Experiments
SandBox3
Experiments
SandBoxSymbolicInverseTrig
Experiments
SandBoxGraphviz
Experiments with GraphViz and StructuredTables
SandBoxDifferentialEquations
Differential Equations etc.
[SandBoxMatrixExample]

[SandBoxRotationMatrix]
Here you can create your own SandBox.
[SandBox9]
Experiments with JET Bundles
[SandBoxGnuDraw]
Miscellaneous
[SandBox11]
Miscellaneous

[[SandBox12TestIndetAndComplex]]

[SandBox13]
Solving some nonlinear differential equations
[SandBox42]
Miscellaneous
[SandBox DoOps]
used to run Axiom without actually have to have it installed!

[SandBoxKMG]
[SandBoxDGE]

[SandBoxMLE]
Maximum likelihood estimation (statistics)
[SandBoxFisher]
Fisher's exact test for 2x2 tables (statistics)
[SandBoxNewtonsMethod]
Newton's method for numerically solving f(x)=0
(with examples of calling Axiom expressions and Spad functions from Lisp).
[SandBoxVeryLongLaTeX]
Test long lines
[SandBox Complementsdalgebrelineaire]
Francois Maltey
[SandBoxFriCAS]
page for testing friCAS
[SandBoxEcfact]
Aldor compiler problem?
[SandBoxMyReduce]
calling reduce with empty list
[SandBoxCategoryTerms]
Category theory terminology used in SPAD

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

Examples
Here is a simple Axiom command:
\begin{axiom}
integrate(1/(a+z^3), z=0..1,"noPole")
\end{axiom}

axiomintegrate(1/(a+z^3), z=0..1,"noPole")
\begin{equation}
\label{eq1}{\left(
\begin{array}{@{}l}
\displaystyle
-{{\sqrt{3}}\ {\log{\left({{3 \ {a^2}\ {{\root{3}\of{a^2}}^2}}+{{\left(-{2 \ {a^3}}+{a^2}\right)}\ {\root{3}\of{a^2}}}+{a^4}-{2 \ {a^3}}}\right)}}}+{2 \ {\sqrt{3}}\ {\log \left({{{\root{3}\of{a^2}}^2}+{2 \  a \ {\root{3}\of{a^2}}}+{a^2}}\right)}}+
\
\
\displaystyle
{{12}\ {\arctan \left({{{2 \ {\sqrt{3}}\ {\root{3}\of{a^2}}}-{a \ {\sqrt{3}}}}\over{3 \  a}}\right)}}+{2 \  \pi}
\end{array}
\right)}\over{{12}\ {\sqrt{3}}\ {\root{3}\of{a^2}}}\end{equation}
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)
And here is a REDUCE command:
\begin{reduce}
int(1/(a+z^3), z,0,1);
\end{reduce}

\begin{reduce}
int(1/(a+z^3), z,0,1);
\end{reduce}

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]?.

... --meliusja,  Tue, 08 Apr 2008 10:33:39 -0700 replyaxiomsolve(a*x+b,x)
\begin{equation*}
\label{eq2}\left[{x = -{b \over a}}\right]?\end{equation*}
Type: List(Equation(Fraction(Polynomial(Integer))))
exploring --Bill Page,  Thu, 24 Apr 2008 07:00:25 -0700 replySandBoxNonAssociativeAlgebra?
lexical scope --Bill Page,  Sun, 11 May 2008 06:43:30 -0700 replyTesting lexical scoping rules in SandBoxLexicalScope?.
Combinatorial Sum --Bill Page,  Fri, 16 May 2008 13:38:20 -0700 replySandBoxSum? (like Product)
Symbolic computations --Bill Page,  Thu, 22 May 2008 12:58:29 -0700 replySandBoxSymbolic?
add inheritance issue --Bill Page,  Sun, 25 May 2008 11:26:13 -0700 replyFor example: SandBoxLeftFreeModule?
Added Preview and Cancel to comment form --page,  Tue, 27 May 2008 15:08:43 -0700 replyThis is a test of the Preview and Cancel buttons:
axiomintegrate(sin x, x)
\begin{equation}
\label{eq3}-{\cos \left({x}\right)}\end{equation}
Type: Union(Expression(Integer),...)
software archeology discovers another --Bill Page,  Fri, 30 May 2008 21:01:37 -0700 replySandBoxSubsetCategory?
try iso-experiment/combinat --Bill Page,  Tue, 03 Jun 2008 13:24:13 -0700 replySandBoxCombinat?
Equation, Inequation, and Inequality --Bill Page,  Mon, 09 Jun 2008 18:28:56 -0700 replySandBoxEquation? SandBoxInequation? SandBoxInequality?
test aldor code --Bill Page,  Wed, 18 Jun 2008 03:43:05 -0700 replySandBoxAdjacencyMatrix?
gnuplottex --Bill Page,  Tue, 24 Jun 2008 22:42:51 -0700 replySandBoxGnuPlotTex?
implementing Integer from Cardinal (unsigned) numbers --Bill Page,  Mon, 21 Jul 2008 06:34:23 -0700 replySandBoxDefineInteger?
Attributes and categories --Bill Page,  Fri, 25 Jul 2008 13:36:54 -0700 replySandBoxCommutativeCategory?
Literal and Symbol in SPAD --Bill Page,  Sun, 27 Jul 2008 02:09:57 -0700 replySandBoxLiteral?
a category of partially ordered sets --Bill Page,  Wed, 06 Aug 2008 17:26:50 -0700 replySandBoxPartiallyOrderedSet?
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[SandBox/interp/i-funsel.boot]?
Tensor Product of Polynomials --Bill Page,  Sun, 24 Aug 2008 06:36:20 -0700 replySandBoxTensorProductPolynomial?
Reflection --Bill Page,  Mon, 08 Sep 2008 06:09:27 -0700 replyspad)abbrev package REFL Reflect
Reflect(T:Type): with
constructor? : Symbol -> Boolean
constructor?(p:Symbol):Boolean == car(devaluate(T)$Lisp)$SExpression = convert(p)$SExpression spad Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2684009892188271010-25px004.spad using old system compiler. REFL abbreviates package Reflect ------------------------------------------------------------------------ initializing NRLIB REFL for Reflect compiling into NRLIB REFL compiling exported constructor? : Symbol -> Boolean Time: 0 SEC. (time taken in buildFunctor: 0) ;;; *** |Reflect| REDEFINED ;;; *** |Reflect| REDEFINED Time: 0 SEC. Cumulative Statistics for Constructor Reflect Time: 0 seconds finalizing NRLIB REFL Processing Reflect for Browser database: --->-->Reflect((constructor? ((Boolean) (Symbol)))): Not documented!!!! --->-->Reflect(constructor): Not documented!!!! --->-->Reflect(): Missing Description ; compiling file "/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/REFL.NRLIB/REFL.lsp" (written 11 JAN 2013 08:24:20 AM): ; /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/REFL.NRLIB/REFL.fasl written ; compilation finished in 0:00:03.044 ------------------------------------------------------------------------ Reflect is now explicitly exposed in frame initial Reflect will be automatically loaded when needed from /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/REFL.NRLIB/REFL >> System error: The bounding indices 163 and 162 are bad for a sequence of length 162. See also: The ANSI Standard, Glossary entry for "bounding index designator" The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR axiomT1:=Integer \begin{equation} \label{eq4}\hbox{\axiomType{Integer}\ }\end{equation} Type: Type axiomT2:=Polynomial Fraction T1 \begin{equation} \label{eq5}\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }))\end{equation} Type: Type axiomT3:=Complex T2 \begin{equation} \label{eq6}\hbox{\axiomType{Complex}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ })))\end{equation} Type: Type axiomconstructor?('Polynomial)$Reflect(T1)
\begin{equation}
\label{eq7} \mbox{\rm false} \end{equation}
Type: Boolean
axiomconstructor?('Polynomial)$Reflect(T2) \begin{equation} \label{eq8} \mbox{\rm true} \end{equation} Type: Boolean axiomconstructor?('Polynomial)$Reflect(T3)
\begin{equation}
\label{eq9} \mbox{\rm false} \end{equation}
Type: Boolean
axiomconstructor?('Complex)$Reflect(T3) \begin{equation} \label{eq10} \mbox{\rm true} \end{equation} Type: Boolean Francois Maltey --Bill Page, Tue, 18 Nov 2008 19:11:35 -0800 replySandBoxConditionalFunctions? NonZeroInteger? --Bill Page, Tue, 02 Dec 2008 21:35:49 -0800 replySandBoxNonZeroInteger? 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 replySandboxDelay? Ralf Hemmecke's example for hidden overloading --Bill Page, Mon, 23 Mar 2009 07:22:32 -0700 replySandBoxHiddenOverloading? prototype for tensor products --Bill Page, Tue, 12 May 2009 12:49:48 -0700 replySandBoxTensorProduct? by Franz Lehner Riemann Surface --Bill Page, Sat, 20 Jun 2009 13:15:37 -0700 replySandBoxComplexManifold? Differential Algebra --Bill Page, Wed, 29 Jul 2009 09:07:26 -0700 replySandBoxDifferentialPolynomial? Grassmann Algebra --Bill Page, Thu, 10 Sep 2009 09:05:28 -0700 replySandBoxGrassmannIsometry? - 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 replySandBoxFreeProduct? 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 replyFranz 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 replyMortonCode? (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 replySandBoxGroebnerBasis? examples from Ideals, Varieties, and Algorithms Third Edition, 2007 Frobenius Algebra --Bill Page, Fri, 11 Feb 2011 17:12:43 -0800 replyFrobeniusAlgebraVectorSpacesAndPolynomialIdeals? 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 replySandBoxObserverAsIdempotent? Some or all expressions may not have rendered properly, because REDUCE returned the following error:Error: export PATH=/usr/local/bin:$PATH; ulimit -t 240; reduce < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7875621063227459889-25px.run


Some or all expressions may not have rendered properly, because Latex returned the following error:
sage: unset PYTHONPATH; PATH=/usr/local/bin:\$PATH HOME=/var/lib/zope sage 749854144765306254-16.0px.sage
/usr/local/bin/sage: line 36: cd: /home/page/sage-3.4-linux-Debian_GNU_Linux_4.0_etch-x86_64-Linux: No such file or directory
/usr/local/bin/sage: line 41: /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/local/bin/sage-sage: No such file or directory