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

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?

FriCASEmacsMode

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

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 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]?

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

SolvingDifferentialEquations

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

SandBoxRealSpace

Some tests to mimic R^n

SandBoxProp

First order language over comparable types (tests for qel)

SandBoxGeom1

Cells and k-surfaces (preps for manifold<-charts)

SandBoxPQTY

Some tests for pqty buckingham pi (physical quantities)

SandBoxTensorProduct3

Tensor product of three different spaces: U#V#W

SandBoxSurfaceComplex

Some tests for k-cells and k-surfaces (Rudin/PMMA)

SandBoxUnify

Some unification tests

SandBoxJacobiDiagFloat

Jacobi diagonalization algorithm

SandBoxJacobiDiagIntervalFloat

Jacobi diagonalization algorithm (Interval Float)

SandBoxPhysicalUnitSystem

Physical unit systems (FreeAbelianGroup? replaced)

SandBoxHyperGeometric

Book A=B

SandBoxLinearProgramming

Revised simplex method

SandBoxDOPT

Discrete optimal transport

SandBoxJacobiDiag

Jacobi diagonalisation

SandBoxRootOfUnity

Roots of unity, https://en.wikipedia.org/wiki/Root_of_unity

SandBoxDFT

Discrete Fourier Transform

SandBoxSEXPM

SExpression? pattern matching (unifier)

SandBoxSSPM

SuperSimplePatternMatcher? (http://www.cs.northwestern.edu/academics/courses/325/readings/pat-match.php)

SandBoxFORMAN

Interactive Computer Manipulation of Formal Sums (http://www.csd.uwo.ca/~watt/home/students/theses/NPatil2010-msc.pdf)

SandBoxDFORM

Differential forms (graded)

SandBoxINFSUM

Test SBCL 1.1 (table matching/lookup using unification)

SandBoxTensorPower

Demo Issues (n:PI ???)

SandBoxTensorAlgebra

Idea: use GradedAlgebra? and TensorPower? without n.

SandBoxDemoFreeMonoid/-Module

Combine monoid/module for algebra like impls.

SandBoxDemoXFreeAlgebra

Demo XDistributedPolynomial(B,R) (see above)

SandBoxTensorAlgebra2

Least effort solution using XFreeAlgebra?

SandBoxTensorAlgebra3

Simplified; -GradedAlgebra?; +Ring (any)

new

SandBox

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.

Examples

Here is a simple Axiom command:

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

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

(1)

And here is a REDUCE command:

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

load_package sfgamma;

load_package defint;

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

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]?.

solve(a*x+b,x)

(2)

integrate(sin x, x)

(3)

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.

)abbrev package REFL Reflect
Reflect(T:Type): with
    constructor? : Symbol -> Boolean
  == add
    constructor?(p:Symbol):Boolean == car(devaluate(T)$Lisp)$SExpression = convert(p)$SExpression

   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): Not documented!!!!
--->-->Reflect((constructor? ((Boolean) (Symbol)))): Not documented!!!!
--->-->Reflect(): Missing Description
; compiling file "/var/aw/var/LatexWiki/REFL.NRLIB/REFL.lsp" (written 31 OCT 2024 09:59:58 PM):

; wrote /var/aw/var/LatexWiki/REFL.NRLIB/REFL.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
   Reflect is now explicitly exposed in frame initial 
   Reflect will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/REFL.NRLIB/REFL

T1:=Integer

(4)

T2:=Polynomial Fraction T1

(5)

T3:=Complex T2

(6)

constructor?('Polynomial)$Reflect(T1)

(7)

constructor?('Polynomial)$Reflect(T2)

(8)

constructor?('Polynomial)$Reflect(T3)

(9)

constructor?('Complex)$Reflect(T3)

(10)

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

Sandbox with some simple Algebra [SimplifyingAlgebraicExpressions]?

g:=1/(x*(1-a*(1-x)))

(11)

integrate(g,x)

(12)

Sympy

(13)

Mathematica

(14)

)set output algebra on
 

)set output tex off

r1:=(16*x^14-125*x^10+150*x^6+375*x^2)/(256*x^16+480*x^12+1025*x^8+750*x^4 +625)

                 14        10        6        2
             16 x   - 125 x   + 150 x  + 375 x
   (13)  ------------------------------------------
              16        12         8        4
         256 x   + 480 x   + 1025 x  + 750 x  + 625

r2:=integrate(r1,x=0..1);

numeric r2

   (15)  0.1239829519_8500641246

unparse(simplify(r2)::InputForm)

   (16)
  "((-77)*((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)*log(((400*5^(1/2)+5000)*((5
  3*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)+(20*5^(1/2)+200))/(5^(1/2)))+(77*(((-5
  3)*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)*log(((16400*5^(1/2)+37000)*(((-53)*5^
  (1/2)+124)/(106480*5^(1/2)))^(1/2)+(70*5^(1/2)+200))/(5^(1/2)))+((-77)*(((-53
  )*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)*log((((-16400)*5^(1/2)+(-37000))*(((-5
  3)*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)+(70*5^(1/2)+200))/(5^(1/2)))+(77*((53
  *5^(1/2)+124)/(106480*5^(1/2)))^(1/2)*log((((-400)*5^(1/2)+(-5000))*((53*5^(1
  /2)+124)/(106480*5^(1/2)))^(1/2)+(20*5^(1/2)+200))/(5^(1/2)))+((-308)*(((-53)
  *5^(1/2)+124)/(106480*5^(1/2)))^(1/2)*atan(((105*5^(1/2)+(-200))*((53*5^(1/2)
  +124)/(106480*5^(1/2)))^(1/2)+5^(1/2))/((105*5^(1/2)+200)*(((-53)*5^(1/2)+124
  )/(106480*5^(1/2)))^(1/2)))+(308*(((-53)*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)
  *atan(((105*5^(1/2)+(-200))*((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)+(-1)*5^
  (1/2))/((105*5^(1/2)+200)*(((-53)*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)))+(308
  *((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)*atan(((105*5^(1/2)+200)*(((-53)*5^
  (1/2)+124)/(106480*5^(1/2)))^(1/2)+5^(1/2))/((105*5^(1/2)+(-200))*((53*5^(1/2
  )+124)/(106480*5^(1/2)))^(1/2)))+((-308)*((53*5^(1/2)+124)/(106480*5^(1/2)))^
  (1/2)*atan(((105*5^(1/2)+200)*(((-53)*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)+(-
  1)*5^(1/2))/((105*5^(1/2)+(-200))*((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)))
  +10))))))))/154"

r3:=integrate(r1,x);

unparse(simplify(r3)::InputForm)

   (18)
  "(((-176)*x^8+(-165)*x^4+(-275))*((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)*lo
  g((420*x*5^(1/2)+(-800)*x)*((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)+(2*x^2*5
  ^(1/2)+5))+((176*x^8+165*x^4+275)*(((-53)*5^(1/2)+124)/(106480*5^(1/2)))^(1/2
  )*log((420*x*5^(1/2)+800*x)*(((-53)*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)+(2*x
  ^2*5^(1/2)+5))+(((-176)*x^8+(-165)*x^4+(-275))*(((-53)*5^(1/2)+124)/(106480*5
  ^(1/2)))^(1/2)*log(((-420)*x*5^(1/2)+(-800)*x)*(((-53)*5^(1/2)+124)/(106480*5
  ^(1/2)))^(1/2)+(2*x^2*5^(1/2)+5))+((176*x^8+165*x^4+275)*((53*5^(1/2)+124)/(1
  06480*5^(1/2)))^(1/2)*log(((-420)*x*5^(1/2)+800*x)*((53*5^(1/2)+124)/(106480*
  5^(1/2)))^(1/2)+(2*x^2*5^(1/2)+5))+(((-352)*x^8+(-330)*x^4+(-550))*(((-53)*5^
  (1/2)+124)/(106480*5^(1/2)))^(1/2)*atan(((105*5^(1/2)+(-200))*((53*5^(1/2)+12
  4)/(106480*5^(1/2)))^(1/2)+x*5^(1/2))/((105*5^(1/2)+200)*(((-53)*5^(1/2)+124)
  /(106480*5^(1/2)))^(1/2)))+((352*x^8+330*x^4+550)*(((-53)*5^(1/2)+124)/(10648
  0*5^(1/2)))^(1/2)*atan(((105*5^(1/2)+(-200))*((53*5^(1/2)+124)/(106480*5^(1/2
  )))^(1/2)+(-1)*x*5^(1/2))/((105*5^(1/2)+200)*(((-53)*5^(1/2)+124)/(106480*5^(
  1/2)))^(1/2)))+((352*x^8+330*x^4+550)*((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/
  2)*atan(((105*5^(1/2)+200)*(((-53)*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)+x*5^(
  1/2))/((105*5^(1/2)+(-200))*((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)))+(((-3
  52)*x^8+(-330)*x^4+(-550))*((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)*atan(((1
  05*5^(1/2)+200)*(((-53)*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)+(-1)*x*5^(1/2))/
  ((105*5^(1/2)+(-200))*((53*5^(1/2)+124)/(106480*5^(1/2)))^(1/2)))+(5*x^7+35*x
  ^3)))))))))/(176*x^8+165*x^4+275)"

r4:=D(r3,x);

--simplify(r1-r4)
--normalize(r1-r4)
r5:=eval(r1-r4,x=10);

numeric r5

   (21)  0.0

)set output algebra on

sin(x^b)

              b
   (22)  sin(x )

D(%,x)

            b - 1     b
   (23)  b x     cos(x )

D(%,x)

            2  b - 1 2     b      2      b - 2     b
   (24)  - b (x     ) sin(x ) + (b  - b)x     cos(x )

)set output algebra off
 

)set output tex on

guessRec([1,1,0,1,- 1,2,- 1,5,- 4,29,- 13,854,- 685])

(15)

integral(x^2, x)

(16)

integrate(x^2, x)

(17)

integrate(log(m*x+b),x)

(18)

integrate(log(m*x+b) * exp(-(log(x) - mu)^2 / (2*sigma^2)) / (X*sigma*sqrt(2*pi)), x)

(19)

integrate(log(b) * exp(-(log(x) - mu)^2 / (2*sigma^2)) / (X*sigma*sqrt(2*pi)), x)

(20)