
A glossary generated from the pages on this site.
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
A  AWAIC
 \begin{spad}
 Disclaimer
 The National Institute of Standards and Technology (NIST) uses its best
 efforts to deliver a high quality copy of the Database and to verify that
 the data ...
 AboutMaxima
 Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of ...
 AboutReduce
 REDUCE: Overview
 AboutSage
 SAGE is free and open software that supports research and teaching in algebra, geometry, number theory, cryptography, etc. Both the SAGE development model and the technology in SAGE itself is ...
 AbstractFauser2007
 Calculations in character Hopf algebras using SCHUR and Maple
 Aldor
 Aldor is a computer programming language, like Scheme, Java or C#.
 AldorFibonacci
 And below an Aldor version of the Fibonacci function that resembles a definition done in Haskell.
 AldorForFriCAS
 It is possible to use the Aldor compiler WebSite:www.aldor.org
to compile functions which you then can use from FriCAS. Currently
the following commands work and install Aldor in '/opt/aldor':
 AldorInEmacs
 There are currently two packages available for emacs which support the "Aldor":http://www.aldor.org
programming language.This page provides information and links to the available tools.
 Award Of Bounties
 **Bounties:** 
relatively small promotional awards to be paid for programming
work done to enhance Axiom.
 AxServer
 \documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra axserver.spad}
\author{Arthur C. Ralfs}
\maketitle
\begin{abstract}
The AxiomServer package is designed to ...
 Axiom Algebra
 In FriCAS adding a new constructor (category, domain or package) is easy, just add
it and update algebra Makefile.in as indicated in note at top of Makefile.in
 Axiom Strengths
 See FriCASAdvantages. The page below is mostly historical.
 Axiom Volume 1
 Axiom Developers,Axiom Volume 1: Tutorial on Lulu.com Here
is a link to the first volume of the new Axiom book on Lulu.com.
Thanks Tim!Bill Page
 Axiom and Maple
 From: http://history.siam.org/pdfs2/Gonnet_final.pdf
 Axiom meeting 2005
 See also: [Next Axiom Meeting]
 AxiomBook
 This is historical entry. For up to date version go to
FriCASBook.
 AxiomCompiler
 See [FriCAS compiler]. The text below is mostly of historical interest.
The language used for programming Axiom is called SPAD. In fact, SPAD has a
cousin, originally intended as a successor called "...
 AxiomDocumentation
 The Axiom Book
 AxiomProgrammingDiscussion
 Metaclasses versus Categories
 AxiomPromotion
 1 Communities to inform of a release
 AxiomScreenCast
 Flash Presentations
 AxiomSources
 For uptodate information about Axiom see http://axiomdeveloper.org
 AxiomUI
 by [Kai Kaminski]
 AxiomWindowsInstallerScript
 \documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\textbf{Axiom Windows Installer Script}}
\author{Jose Alfredo Perez}
\maketitle
B  BNatural
 A new project has been proposed to attempt to implement BNatural
using Aldor as an extension of the open source version of Axiom.
 BeBold
 Follow the Wikipedia guideline of
"being bold":http://en.wikipedia.org/wiki/Wikipedia:Be_bold
in updating wiki pages: Don't ask, just update (i.e. improve) the page.
 BiCartesianTensor
 BiCartesianTensor(dim,R) provides Cartesian tensors with
components belonging to a commutative ring R. These tensors
can have any number of covariant and contravariant indices.
Each index takes ...
 BootProgramming
 Boot (more precisely newer dialect called Shoe) is used internally in FriCAS for some of the interpreter
and compiler code. Here is a detailed description of the
"Boot Language":/axiomtest1/src/...
 BugTests
 This page serves as parent for pages testing possibly buggy behaviour.
 BugTracker
 FriCAS bugs should be reported in FriCAS bug tracker at SourceForge:
http://sourceforge.net/p/fricas/bugs/
 BuildFriCAS
 Building on Linux
 BuildOnCygwin
 Note: the text below should be identical to the contents of INSTALL.CYGWIN in
the FriCAS distribution.
 BuildOnWindows
 Building FriCAS on Windows
 BuildOpenAxiom
 Building on Linux
C  CaleyDickson
 Ref:
 Cartesian Product
 This domain implements cartesian product, we give example usage here:
 Category Theory and FriCAS
 ClosedLinearOperator
 Introduction
 Common Mistakes
 1 Omitting the {axiom} enviroment
 CommonDenominator for polynomials
 This page presented extension of UnivariatePolynomialCommonDenominator for arbitrary polynomial categories. FriCAS now
contains such extension, so we just present an example.
 Complex Polynomial
 Why this evalutate to zero ?
\begin{axiom}
A: Complex Polynomial Integer
A*conjugate A  A^2
\end{axiom}
Definitely wrong. It looks like the source of the problem is:
\begin{axiom}
conjugate(a)
\...
 Computing with Vectors
 How to multiply two vectors??
 Constant of integration
 By definition definite integral is determined only up to integration constant.
When integral depends on parameters constant of integration also depends
on parameters. Some choices of integration ...
 CopyrightPage
 Axiom was released under this license as of September 3, 2002.
 Creating New Pages
 Suppose someone asks you: 
**I think it would be great if you could add your
algoritm to the FriCASWiki web site.**
 Cylindrical Algebraic Decomposition
 Note: This package is currently included in FriCAS.
D  DancingSambaRamanujan
 Dancing Samba with Ramanujan Partition Congruences
 DebuggingFriCAS
 Fortunately, FriCAS is rather easy to debug. Most of the source is very transparent, so once you have localized the bug there are mostly "design issues":DesignIssues to overcome. However, to find out ...
 DedekindEtaFunctions
 Some links related to Dedekind Eta Functions
 Definite Integration
 Here are some integration problems.
 DependentTypeTest1
 \begin{spad}
)abbrev category ACAT A_Categeroy
A_Categeroy() : Category == SetCategory with
degree : % > Polynomial Integer
)abbrev domain BDOM B
B(p) : Exports == Implementation where
...
 DependentTypeTest2
 \begin{spad}
)lisp (defvar $inclAssertions nil)
 DerivFunc
 \begin{axiom}
 Design By Contract
 Software and especially hardware frequently is treated as "given". More precisely,
hardware and software has properties and users depend on its properties. Some properties
are documented, some ...
 DesignIssues
 This should become a page where we document design issues.
 DistributedExpression
 Francois Maltey (FM) and Bill Page (BP) wrote::
 Division by zero during evaluation
 Evaluation may lead to spurious division by zero. This problem is not limited
to FriCAS, in fact it is general mathematical fact which limits what computers
(and people) can do. Simplest example is ...
 DocBook
 http://www.docbook.org/
 Doyen @ ASEE MidAtlantic Spring Conference 2006
 Poster session showing the Doyen Project at New York City Tech. This is the website of the conference: http://websupport1.citytech.cuny.edu/Faculty/hli/asee/asee/default.asp .
 DoyenCD
 Doyen Live CD
 DoyenDocs
 \documentclass{article}
\usepackage{axiom}
\usepackage{graphics}
\begin{document}
\title{Doyen Build Instructions}
\author{Tim Daly \and Jose Alfredo Perez \and Wilken Rivera}
\maketitle
\begin{...
 DoyenFeedBack
 DoyenVirtualMachine
 Following Mr. Tim Daly's suggestion, I have uploaded a virtual machine configuration file for the DoyenCD using VMWare. This should allow to run the DoyenCD in different operating systems, ...
 Dynamic Function Domains
 On Wednesday, January 12, 2005 12:00 PM you wrote:
Ralf Hemmecke wrote:
Well, I haven't checked whether it should really work, but
shouldn't be::
Foo: with {
g: (n: PositiveInteger, k: ...
 dirichlet.spad
 Drichlet ring is now included in FriCAS
 doyen
 Doyen
The {problem, need, market} seems to be to develop a scientific
computation platform. The goal of such a platform would be to
create an ecosystem where scientists can
 develop ...
E  ElementaryFunctionStructurePackage example
 In FriCAS expression are represented as rational functions in kernels.
exp(2*x) and exp(x) are different kernels, so
\begin{axiom}
exp(2*x)  exp(x)^2
\end{axiom}
gives nonzero result. To recognize ...
 ElementaryIntegrationExample
 \begin{axiom}
)set output tex off
)set output algebra on
integrate(x/sqrt(x^4 + 10*x^2  96*x  71),x)
\end{axiom}
 EtaRelations8
 Relations among eta functions of level 8
 ExampleCharacterTable
 We want to compute characters of wreath product Z_3^(Z_3)*Z_3
 ExampleCompositeFunction
 In practice we frequently need to deal with composite functions.
This is frequently formulated as 'y' depends in 'x' where 'x'
is a function (say of argument 't'). Unlike some other systems
in ...
 ExampleFreeAbelianGroup
 How to use FreeAbelianGroup
 ExampleGroebner
 Application of Groebner Bases
 ExampleInequalitiesViaCAD
 Recenty in sci.math.symbolic there was a question if
\begin{equation}
b^2 \geq (a  c)^2
\end{equation}
and
\begin{equation}
a^2 \geq b^2
\end{equation}
implies
\begin{equation}
c^2 \geq (a + b)^2
\...
 ExampleIntegration
 \begin{axiom}
f := x/(x^3x+1)
)set output tex off
)set output algebra on
i := integrate(f, x)
)set output tex on
)set output algebra off
D(i, x)
 ExampleSkewPolynomial
 Note: 04Apr2021: This page is abandoned. It's new place is
https://fricas.github.io/fricasnotebooks/FriCASSkewPolynomial.html
 ExampleSolve
 This is motivated by discussion in Sage mailing list. We are given sum of ellipsis and
a strip symmetric with respect to one of the axis of the ellipsis. We want to
find circle with center on the ...
 ExampleSolve2
 The following example appeared in Maxima mailing list (Maxima could
not do it). Here we solve it using FriCAS. First we form
system of equations:
\begin{axiom}
 Form system of equations
eqn1 := ...
 ExampleXDistributedPolynomial
 Demonstration of XDistributedPolynomial
 ExpressionInteger
 This page gives a few comments about the expression domain 'Expression Integer'
and the use of more specific domains.
 editor
 Just placeholder
F  FAQ
 Frequently Asked Questions
 FRICASsys
 is the main FriCAS program. It does not include graphics or
hyperdoc browsing. Under Linux FriCAS is usually started
via a script called 'fricas' that also starts other processes
such as sman (the ...
 FRISCO
 This is just a link to a european project FRISCO in which software in Axiom and Aldor was developed.
 FactorizationExample
 The following example appeared on Maxima list. Maxima needs 3.5 GB memory and
almost a hour to do it. In FriCAS it is much faster:
 ForksInfo
 In 2007 open source Axiom project forked twice, leading to three
separate projects (forks):
 FriCAS is a fork of the original open source Axiom project created by Waldek Hebisch on July 9, 2007 ...
 FormalFraction
 Author  M.G. Richardson
 Found Objects
 If you have a lot of spare time, you might want to classify those pages, accidentally stumbled across:
 FractionFreeFastGaussianElimination
 FractionFreeFastGaussianElimination implements an algorithm proposed by Bernhard Beckermann and George Labahn. It can be used to solve general rational interpolation problems like (usual) linerized ...
 Free Aldor
 Aldor is now free. Most of material below is only of historical interest.
 FreeModule
 FreeModule implements free module
over a ring with generators indexed by a set.
Each element can be expressed as a finite linear combination of
generators. Only nonzero terms are stored.
 FreeModuleCategory
 A domain of this category implements formal linear combinations
of elements from a domain **S** with coefficients
in a domain **R**. The coefficient ring
may be noncommutative.
 FreeMonoid
 The free monoid on a set S is the monoid of finite products of
the form reduce(*, [si ^ ni]) where the si's are in S, and the ni's
are nonnegative integers. The multiplication is not commutative.
...
 FreeRing
 \begin{axiom}
)lib FPROD
)lib FSUM
\end{axiom}
 FriCAS
 "!FriCAS Project":http://sourceforge.net/svn/?group_id=200168
 FriCAS Algebra
 In FriCAS the word 'algebra' normally means collection of constructors
(categories, domains and packages) provided with FriCAS. For this meaning
see [FriCAS Library]. The source code
for algebra is ...
 FriCAS Binaries
 You can find Fricas installation files at http://fricas.sourceforge.net/download.html
 FriCAS Book
 FriCAS Book is up to date version of Axiom Book by Jenks and Sutor. Master sources are bundled
with FriCAS. HyperDoc shows it when you choose 'Reference' and then 'FriCAS Book'.
Up to date '.pdf' ...
 FriCAS Colloquium
 The Colloquium's purpose is twofold: 1) for the participants to share and discuss theory, practical experiences and insights regarding FriCAS, and 2) to assist others concerned with FriCAS and ...
 FriCAS Library
 The [FriCAS Library] contains over 1,000 stronglytyped mathematical domains
and categories. It is written in "Spad":FriCASCompiler. The source code for
the library is here:
 FriCAS Library Examples
 Here we collect some examples of using FriCAS library
 FriCAS Problems
  [Unevaluated Expressions]
 FriCAS Sources
 Links to release tarballs  http://fricas.sourceforge.net/download.html
 FriCAS Syntax
 On January 19, 2006 5:53 AM Fabio Stumbo wrote:
 FriCAS compiler
 The language used for programming FriCAS is called Spad. Currently most detailed
and up to date documentation is in FriCAS book at http://fricas.github.io/book.pdf There
are changes compared to ...
 FriCAS language
 FriCAS has the [FriCAS interpreter] for interaction with the
user and the [FriCAS compiler] for building library modules.
 FriCASAdvantages
 Compared to old Axiom FriCAS has following advantages:
 FriCASContributions
 The following are not included in current FriCAS:
 FriCASDocumentation
 FriCAS interface documentation
 FriCASEmacsMode
 FriCAS contains upto date Emacs mode. You can start FriCAS inside Emacs typing 'efricas'.
 FriCASGraphics
 Graphics in FriCAS
 FriCASHelpSet
 )help set
 FriCASIcon
 Some Proposed Site Icons
 FriCASInVim
 Syntax highlighting for FriCASfiles: .spad, .input
 FriCASIntegration
 FriCAS inherited from Axiom implementation of Risch algorithm for elementary integration. It can
handle both easy cases
\begin{axiom}
integrate(x*exp(x^2), x)
integrate(exp(1/x^2)/x^3, x)
\end{axiom}...
 FriCASInterpreter
 The FriCAS interpreter is the part of FriCAS responsible for handling
user input during an interactive session.
 FriCASIssues
 I have a number of issues that are somewhere between bug reports, new feature requests, problem workarounds and so on. Perhaps if I start to write them down some structure will appear. The issues on ...
 FriCASModifications
 Around August 2011 there have been made some changes that are described concisely in a Mail by Waldek:
http://groups.google.com/group/fricasdevel/msg/1d069ca72125d17c?hl=en
 FriCASOutputFormatting
 One of the most frustrating things as a novice FriCAS user is to try to figure out how to get
FriCAS output to appear in the desired form. For instance:
 FriCASPreGenerated
 This is a description of how to generate and how to use the new
'pregenerated' system.
 FriCASProgramming
 General programming
 FriCASProgrammingModel
 FriCAS language (both interpreter and Spad) is imperative, basic operation is an assignment like
\begin{axiom}
a : Integer := 5
b : Integer := a + 2
\end{axiom}
FriCAS language is typed, each ...
 FriCASSpecialIntegration
 FriCAS can now handle large class of integrals expressible in terms of exponential
integral, error functions, incomplete Gamma function with constant
first argument, logarithmic integral and ...
 FriCASTimofeev1
 Examples from Timofeev (1948) Ch. 1, divided into sections
kindly provided by clicliclic vel Martin on sci.math.symbolic
 FriCASTimofeev37
 Examples from Timofeev Chapter 3 and Chapter 7
 FriCASTimofeev5
 Examples from Timofeev (1948) Ch. 5, divided into sections
kindly provided by clicliclic vel Martin on sci.math.symbolic
 FriCASTimofeev9
 Timofeev Chapter 9 Integration Problems
 FriCASTutorial
 Most introductory material about Axiom applies to FriCAS.
The first chapters of FriCAS book http://fricas.github.io/book.pdf
may serve as a tutorial.
 FriCASWiki
 The FriCAS Wiki (this web site) is an online enviroment in which
FriCAS users and developers can:
 FriCASonCygwin
 Martin Rubey provides the following instructions for running FriCAS on Cygwin:
 Frobenius Algebra, Vector Spaces and Polynomial Ideals
 References
 FrontPage
 [News]
 FrontPageDiscussion
 Please add comments about FrontPage here...
 FunWithFunctions
 \begin{spad}
)abbrev package MYEXP MyExp
 FunctionalProgramming
 In pure functional programing there are no variables, there are only values. Values
are immutable and logically exist "forever" from creation. Program creates
new values from old ones, typically ...
 finite field expressions
 Here are some notes I took when I tried to allow expressions ove finite fields.
 fricasdevel
 The FriCAS developer email list is here:
G  GraphViz
 "An Introduction to GraphViz":http://www.linuxjournal.com/article/7275
 Graphics on MathAction
 Graphics?
 GroupAlgorithms
 FriCAS implements several algorithms for working with groups:
 Guessing formulas for sequences
 Author: "Martin Rubey":http://www.mat.univie.ac.at/~rubey/martin.html
H  HeapSort1
 Heapsort
 HelpPage
 "What's going on here?":#whatisthis
"User interface":#ui
"Navigation":#navigation
"Subscription":#email
"Editing":#editing
"Formatting rules in a nutshell":#rules
"Finding out more":#more
 HistoricAxiom
 History
 HyperDoc
 FriCAS in X11 environment (for example Linux) includes graphics and working HyperDoc browser.
 HyperDocTracing
 The Trace Facility
 HyperG
 \begin{spad}
)abbrev package SPI SumProdInt
SumProdInt(R): Exports == Implementation where
R:Join(Comparable,IntegralDomain,RTZZ,LEZZ)
I  Imitation As Complement
 MaplePrimes
 Indefinite Types
 Declaring Types
The idea that one should be able to "declare the type" of a
variable in FriCAS by the command
\begin{axiom}
n:PositiveInteger
\end{axiom}
 IndexedUnion
 Goal: To explore the issues and develop tools for implementing 'IndexedUnion', a new domain constructor proposed by William Sit.
 InputForm
 Domain of parsed forms which can be passed to the interpreter. This is also the interface between algebra code and facilities in the interpreter.
 Installation Files
 PRECOMPILED BINARIES
 Interval Arithmetic
 Interval arithmetic for Axiom was implemented by Mike Dewar in form of Aldor files.
Juergen Weiss converted the files to Spad and they are included in current FriCAS
distribution.
 IssueTracker
 Bug reports and other issues.
Please identify yourself and provide an email address using
login
before editing or adding a new issue.
See also:
! ...
 indexed variables
 I am trying to create formulae for products with, say, p sub k as
symbols. These are the results from two, at least naively identical formulae:
J  JenksSutorInXhtml
 This is the JenksSutor book in xhtml + mathml.
The graphics are still included as images in the ps.tar
file. The ps directory with the pictures and the graphicstyle.css file
should be
in the same ...
 Jet Bundles
 Note: In 2011 JET bundle package was included in FriCAS.
 Jet Figure 4
 \begin{axiom}
jb := JBUNDLE(['x,'y,'z],['u]);
jbx := JBX jb;
jbl := JBLF(jb,jbx);
de := JDE(jb,jbl);
ck := CKP(jb,jbl);
\end{axiom}
 Jet Figure 5
 \begin{axiom}
jb := IJB('x,'u,'p,4,3);
jbe := JBE jb;
de := JDE(jb,jbe);
ck := CKP(jb,jbe);
\end{axiom}
 Jet LUDecomposition
 !LUDecomposition (LUD)
 jEdit
 Outline
K  Kai Kaminski
 Biography
L  LaTeX
 LaTeX ... is the world standard for writing technical documents. Authors in
science and engineering are probably already familiar with it (or should be).
If you've wrestled with and cursed at "...
 LanguageDifferences
 As mentioned on FriCASProgramming, there are a number of languages connected to PanAxiom.
Since for actual users only 3 of them are relevant, namely,
"Spad":FriCASCompiler,
the "FriCAS interpreter":...
 LatexSymbols
 \documentclass{article}
%%% ====================================================================
%%% @LaTeXfile{
%%% filename = "symbols.tex",
%%% version = "2.00",
%%% ...
 LatexTemplate
 %% This file is used as a python printf() string. As such, the document we are
%% trying to render is placed at the location of the percents below, and any
%% comments in this file must be ...
 LatexWiki
 Test
\begin{equation}
\alpha = \frac{\beta}{\gamma}
\end{equation}
 Limits and Colimits
 Limits
 LineBreaker
 From  Bill Page
 LinearOperator
 Introduction
 LinuxDoc
 http://www.faqs.org/docs/Linuxmini/LinuxdocReference.html
 Lisp's Fraction Integer Domain
 Original Date: Fri, 25 Mar 2005 12:01:44 0600
 LispInFriCAS
 It is possible to enter Lisp expressions directly into FriCAS
(including function definitions) using the ')lisp' command,
but this is rather awkward for long Lisp programs.
\begin{axiom}
)lisp (defun ...
 List To Matrix
 Assume that I have a 'List', like:
\begin{axiom}
L := [[ 1,3, 3,1],[3, 6,3],[ 3,3],[1]]
\end{axiom}
(which is calculated from some earlier expressions).
How can I convert it into a 'SquareMatrix'....
 ListProgramming
 List is a sequence of nodes storing data and links to other node. Each node has place for single piece of
data and a single link. First node of the list contains link to second node, second node ...
 LiterateProgramming
 Note: this is historical info. FriCAS sources are just Spad
files and Axiom changed its format.
 LyX
 !**LyX** is a fully featured document processor, including:
M  Manipulating Expressions
 The domain InputForm can be quite useful for manipulating parts
of expressions. For example
 MathAction
 Welcome to !MathAction  The Interactive Mathematical Web!
 MathAction Problems
 Some special FriCASspecific LaTeX commands might not be understood
by MathAction. For example::
 MathActionAssistantHelp
 The MathAction Assistant is a new interactive toolbar that appears at
the bottom of the 'edit' and 'Add comments' windows. It helps users
edit text containing LaTeX, Axiom and Reduce commands. ...
 MathActionDevelopment
 Design Goals and Issues
 MathML
 http://www.w3.org/Math/
 MathMLFormat
 The MathML package
 Mathaction2Input
 \documentclass{article}
\usepackage{axiom}
\begin{document}
\title{mathaction2input}
\author{Bill Page and David Cyganski}
\date{5 July 2007}
\maketitle
\begin{abstract}
This script converts the ...
 MiKTeX
 MiKTeX is an uptodate TeX implementation for the Windows operating system.
 MinGW
 What is MinGW?
 Mirrors
 .hoverred:hover {backgroundcolor: pink; textdecoration: none }
.hoverred:link {textdecoration: none }
.hoverred:active {textdecoration: none }
.hoverred:visited {textdecoration: none }
....
 MortonCode
 References to Morton codes (also known as zorder and Lebesgue curves):
 Multivariate Resultants
 Here is what I got so far by translating Amit Khetans bires.mpl into Axiom (I'm struggling with guessing the right coercions...) Help is very much appreciated!::
 Mutual Recursion
 Here we give two examples of defining functions by mutual recursion in FriCAS
 MyComplex
 \begin{spad}
)abbrev domain MYPLEX MyComplex
++ Description:
++ Based on \spadtype{Complex(R)},
++ \spadtype{MyComplex(R)} creates the domain of elements of the form
++ \spad{a + b * i} where \spad{a}...
 mathematical algorithms
 Computer algebra systems like FriCAS implement a very large number of
[mathematical algorithms]. By that we mean:
N  Nature of expressions
 FriCAS expression domains are based on notion of differential field. This
have advantages, for example some otherwise unsolvable problems became
solvable when dealing with differential fields. But ...
 News
 Please post notices and information about events, conferences
and meetings related to FriCAS and computer algebra here.
 Next Axiom Meeting
 **On November 19, 2005 10:34 AM Tim Daly (root) wrote:**
 NonCommutativeLaurentPolynomials
 Consider a noncommutative group ring over commuting polynomials
\begin{axiom}
NPOLY:=MonoidRing(Polynomial(Fraction(Integer)),FreeGroup(Symbol))
\end{axiom}
 NoncommutativePolynomials
 There is are some domains in FriCAS for doing computations with
noncommuting variables developed by Michel Petitot. You can find
some examples in the FriCAS book under the title XPolynomial but
...
 NumericalIntegration
 The package 'NumericalQuadrature' implements several operations for numerical integration. Here I provide just some examples. Documentation is to be found in the corresponding '.spad' or '.pamphlet' ...
 NumericalQuadrature
 Numeric integration routines
\begin{axiom}
)sh NumericalQuadrature
\end{axiom}
 new
 begin/{axiom}
[1/7, 3/4, 5/6]
end/{axiom}
 noncommutative Groebner bases
 FriCAS can compute Groebner bases
for noncommutative polynomial rings of solvable
type (of category SolvableSkewPolynomialCategory).
Below we give example using partial differential operators:
\begin{...
 numerical linear algebra
 I want to get (estimates of) the eigenvalues of a 10x10 matrix of floats:
O  Octonion Algebra is Frobenius in Just One Way
 Octonion Algebra Is Frobenius In Just One Way
 OldFrontPage
 **Welcome!**
 OldGnuDraw
 **Demonstrate how graphs may be plotted and displayed from !MathAction**
 OldSummerOfCode
 Summer of Code: Axiom Projects
 OldTeXmacs
 "GNU TeXmacs":http://www.texmacs.org/tmweb/home/welcome.en.html is a free scientific text editor, which was both inspired by TeX and GNU Emacs. The editor allows you to write structured documents via ...
 OldWishList
 This is the old AXIOM wish list, kept for historical reasons.
 OpenAxiom
 Note: Apparently OpenAxiom developement stopped in 2015. Currently there
is no runnable OpenAxiom in the wiki. The rest of the page is just for
historical record.
 OpenAxiom Sources
 Repository  http://www.openaxiom.org/download.html
 OpenAxiomWindows
 Installing OpenAxiom on Windows
 OrderedSet
 The class of totally ordered sets, that is, sets such that for each pair of elements (a, b)
exactly one of the following relations holds a < b or a=b or b < a
and the relation is transitive, i.e. a <...
 OrderedVariableList
 This domain implements ordered variables
\begin{axiom}
)sh OrderedVariableList
\end{axiom}
 OtherComputerAlgebraSystems
 Someday (maybe) we will also include online access to
other systems such as:
 OverloadProblems
 **Type Conversion and Overloading problems**
P  PAFF
 "PAFF":http://wwwrocq.inria.fr/codes/Gaetan.Hache/PAFF.html
: Package for Algebraic Function Fields in one variable
 PanAxiom
 This page explains what PanAxiom stands for.
 PanAxiomCommunity
 Besides frequently visiting the [FriCAS Wiki] website.
There are several ways to stay in contact with the PanAxiom community.
 ParticipantsRISC2007
  Rafal ABLAMOWICZ (Tennessee Technological University)
 PascalTriangle
 Below there is a function which generates Pascal triangle by iteratively filling
rows of a matrix using recurence formula. Spad array indexing is one based,
so we cheat and add 1 to indices.
\begin{...
 PascalTriangleNopile
 \begin{spad}
)nopile
)abbrev package PASCALN PascalNopile
PascalNopile : with {
make_pascal : Integer > Matrix(Integer)
} == add {
make_pascal(N : Integer) : Matrix(Integer) == {
nn := N::...
 Pattern Matching
 Say, I have
\begin{axiom}
test:=D(sin(z)^2, z)
\end{axiom}
 PerCent
 There are some identifiers in FriCAS that start with a percent sign.
 Piecewise Functions
 Consider the following function, given in recursive manner:
\begin{axiom}
N0(t(t1))==0
N0(t(t>=0) and (t0)==(ti)/p*N(t,i,p1)+(i+1t)/p*N(t,i+1,p1)
\end{axiom}
This is a way to create (uniform) ...
 PolyMake
 http://www.math.tuberlin.de/polymake
 Polynomial Coefficients
 Let's examine a simple case:
\begin{axiom}
Dg := [p3  3*p2 + 3*p1  p0,3*p2  6*p1 + 3*p0,3*p1  3*p0,p0]
\end{axiom}
Now calculate coefficients in two ways:
\begin{axiom}
map(coefficients, Dg::...
 PolynomialOverFiniteField
 \begin{axiom}
p ==> 3
K ==> PrimeField p
P ==> UnivariatePolynomial('x, K)
x: P := monomial(1,1)$P
f: P := (x+2)*(2*x^5 + x^3 + 2)
g: P := 2*x^3 + x^2 x +1
f/g
factor f
factor g
k: K := 2
f/k
\end{...
 ProgrammingSPAD
 A very brief introduction to programming in SPAD
Q  QuadraticForm
 \begin{axiom}
)show QuadraticForm
\end{axiom}
R  RandomAlgebra
 (the maths inside is not meant to be taken seriously;
'tis a silly idea that can't work)
 Rational Interpolation
 Introduction
 RealClosure
 This contribution is included in FriCAS. Info below is partly
historical, parlty adds documentation.
 RealNumbers
 FriCAS implements the number domains Integer, Float and even
Fraction(Integer) i.e. rational numbers, but it does not
implement anything called Real numbers. *Why is that?*
 RebuildDoyen
 
Instructions to make a new Doyen image file.

 RecurrenceRelationOperator
 Here is a simple implemention of a recurrence relation operator. It is far
from finished, but might serve as a starting point. I experienced the following
difficulties:
 ReduceAppendixB
 Appendix B Examples
 ReferenceSemantics
 Traditional imperative languages like C, Pascal etc. have value semantic: variables store possibly
complicated compound values. Normal calling sequence copies values of function parameters so
that ...
 Rep and Per
 Aldor programming philosophy from the point of view of it's
evolution from previous incarnations of the Axiom library
compiler, is described here:
 RichExp1
 Examples involving exponential function from Albert Rich testsuite.
 RichExp2
 Examples involving exponential function from Albert Rich testsuite.
 RischImplementationStatus
 In FriCAS the Risch algorithm is split into two stages, a preparatory one
where the integrand is rewritten to express it in terms of differential
fields, and the core part.
 Root denesting
 FriCAS can denest many nested roots. This is implemented in an extra package so
befor use we need to expose it:
\begin{axiom}
)expose RootSimplification
\end{axiom}
 Rosetta
 \documentclass{article}
\normalsize\baselineskip=12pt
\parskip=0pt
\parindent=10pt
\tolerance=5000
\pretolerance=5000
\frenchspacing
\hangindent=10pt
\skip\footins=18pt
\global\textwidth 31pc \global\...
 RosettaStone
 1 Rosetta Translations
 reading structures from a file
 This package provides operations that all take a filename and return an axiom structure which is supposed to be in some standard notation in this file. Geared towards polymake. Needs '...
 realElementary
 \begin{axiom}
realElementary sinh x
realElementary cosh x
\end{axiom}
 robots.txt
 # Define accessrestrictions for robots/spiders
# http://www.robotstxt.org/wc/norobots.html
S  SandBox
 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 ...
 SandBox 1D output Form
 **How to have an output similar to a 1D input form?**
 SandBox Aldor Category Theory
 This project is intended to implement the code and concepts
presented by Saul Youssef in "Prospects for Category Theory
in Aldor", 2004.
 SandBox Aldor Category Theory 3
 \begin{aldor}[slicecategories]
#include "axiom"
#pile
 SandBox Aldor Category Theory 4
 \begin{aldor}[morphisms]
#include "axiom"
#pile
#library lBasics "basics.ao"
import from lBasics
 SandBox Aldor Category Theory 5
 \begin{aldor}[categories]
#include "axiom"
#pile
 SandBox Aldor Category Theory 6
 \begin{aldor}
#include "axiom"
#pile
 SandBox Aldor Category Theory 7
 \begin{aldor}[subobjects]
#pile
#include "axiom"
+++
+++ Quotients
+++
define Quotient(Obj:Category):Category == with
Quotient: (A:Obj,B:Obj) > (A>B) > (Quo:Obj,A>Quo,(X:Obj)>(A>X)>(Quo...
 SandBox Aldor Category Theory Basics
 Miscellaneous Logical helper functions
 SandBox Aldor Category Theory Categories
 \begin{aldor}[categories]
#include "axiom"
#pile
 SandBox Aldor Foreign
 On Wednesday, May 31, 2006 2:54 AM Gernot Hueber wrote:
Due to that I want to call external library functions from within
Aldor/Axiom I did some trials with Aldors "Foreign" and GCL "defentry"
...
 SandBox Aldor Generator
 Aldor has generators:
 SandBox Aldor Join and Meet
 AldorMeet
 SandBox Aldor Semantics
 The following Aldor code was given as an example by Ralf Hemmecke
and added to the wiki by Martin Rubey. See the thread:
 SandBox Aldor Sieve
 From section 1.2 of Aldor Users Guide
 SandBox Arrays
 Adding new functions to PrimitiveArray in an attempt to improve
performance.
 SandBox Boolean
 On Wed, 1 Feb 2006 10:29:24 0700 Robert Dodier asked C Y:
 SandBox Cast
 This is an example of using 'pretend' to perform a type
unsafe operation.
 SandBox Categorical Relativity
 Special relativity without Lorentz transformations.
 SandBox Category of Graphs
 FriCAS currently has a subsystem for*graphs*. The discussion below
is morstly of historical interest. Note that some names in code below
are the same as names in current implementation which
...
 SandBox Category of Graphs 2
 This seems to work if the category is defined in the same source
file (i.e. same section in MathAction). The code is based on the
Spad version [SandBox Category of Graphs in Spad]
\begin{aldor}
#
...
 SandBox Category of Graphs in SPAD
 First we define the general category of graphs.
\begin{spad}
)abbrev category GRAPHS GraphCategory
GraphCategory(nodes:Type, edges:Type): Category == with
source:edges>nodes
target:edges>nodes
 SandBox Direct Product
 The direct product of two functions over the same domain is
another function over that domain whose codomain is the product
of the codomains their codomains.
 SandBox DoOps
 \begin{axiom}
zerosOf(75*x+5*x^2x^3,x)
\end{axiom}
 SandBox Einstein Velocity
 Einstein Velocity and Nonassociative Addition
 SandBox Embeded PDF
 This is how a pamphlet file will look in MathAction
 SandBox EndPaper
 Basic Algebra Hierarchy
 SandBox Experimenting
 \begin{axiom}
 SandBox Folding
 Real Easy Folding
 SandBox Functional Addition
 This library code is based directly on MappingPackage3 in
'mappkg.spad.pamphlet'
 SandBox Functions
 SandBox Gamma
 From Numerical Recipes in C, 2nd edition
 SandBox Grassmann Algebra Is Frobenius In Many Ways
 Grassmann Algebra Is Frobenius In Many Ways
 SandBox Idempotent Observers
 Preliminaries
 SandBox Integration
 Integration
 SandBox LaTeX
 $x^2$
 SandBox Lisp
 Testing Lisp under FriCAS Interpreter
\begin{axiom}
)lisp (setq $dalymode t)
(+ 2 3)
\end{axiom}
 SandBox Lorentz Transformations
 Lorentz transformations relate one object or observer (represented by a
timelike 4vector) to another object or observer. A Lorentz transformation
is **not** uniquely given by the relative velocity ...
 SandBox Manip
 Demonstration of modifying and testing a SPAD file
 SandBox Manipulating Domains
 On Sunday, March 19, 2006 5:54 AM Francois Maltey wrote:
 SandBox Mapping
 This domain provides a functionlike type that can be queried
for domain and codomain.
\begin{aldor}[sig]
#include "axiom.as"
T ==> SetCategory;
 SandBox Matrix
 Symbolic Matrices
 SandBox Monoid
 \begin{spad}
)abbrev category MM MyMonoid
)if false
 Does not work now
Monoid(m:Symbol,u:Symbol): Category == with()
m: (%,%) > % ++ returns the product of x and y
u: () > % ...
 SandBox Monoid Extend
 Martin's beautiful idea:
 SandBox NNI
 This is an experiment to see what the differences are
between defining NonNegativeInteger (NNI) as a SubDomain
of Integer, versus defining it simply based on Integer
itself.
 SandBox Numerical Integration
 On March 10, 2006 10:53 PM Donald J Bindner wrote:
 SandBox Object
 Typesafe Objects in SPAD
 SandBox Polynomials
 Testing a change from::
 SandBox ProblemSolving
 MatheAbi ab 2004
 SandBox Quaternion Algebra is Frobenius in Many Ways
 Quaternion Algebra Is Frobenius In Many Ways
 SandBox Qubic
 \begin{axiom}
digits 20
 n:=x^3+a1*x^2+a2*x+a3 ::Polynomial Fraction Integer
 SandBox Reflection in Aldor
 On July 28, 2006 3:21 AM Christian Aistleitner wrote:
 SandBox Risch
 Let us integrate functions as explained in:
IV_ (https://math.stackexchange.com/users/292527/iv), Does someone know an onlineapplet for the Rischalgorithm?, URL (version: 20200820): https://...
 SandBox SPAD dependent types
 ** Notations **
 SandBox SPAD for Python
 In Python we might write:
 SandBox SPAD for Python 2
 Aldor and SPAD allow domains to have parameters. For example
let's generalize the previous Complex domain to form complex
numbers over any ring.
 SandBox Sedenion Algebra is Frobenius In Just One Way
 Sedenion Algebra is Frobenius in just one way!
 SandBox Set Any
 In Issue #347 it is shown that set equality fails after applying
a map to a set:
\begin{axiom}
A:Set Integer := set [2,1,0]
B:Set Integer := set [0,1,4]
C:=map(x +> x^2,A)
test(C=B)
\end{axiom}
 SandBox Shortcoming
 Please try the two solve statements at the bottom
The first one runs in a short time but
the second runs a long time. The only
differenct is '*R*P' in eq1. Examining eq1a shows that
in fact p=1 has ...
 SandBox Sockets in SPAD
 SocketsInSpad
 SandBox Solve
 Solving Equations
 SandBox Spad
 A minimal exemple of my trouble.
 SandBox Speed
 \begin{aldor}
#include "axiom.as"
 SandBox Statistics
 Testing Martin's statistics code
 SandBox SubDomain
 Exposing the SubDomain constructor:
\begin{spad}
)abbrev domain SUB SubDom
SubDom(X:Type,C:Type):Type with
if C has IntegralDomain then IntegralDomain
== SubDomain(X,true)
\end{spad}
 SandBox Tail Recursion
 On Tuesday, August 30, 2005 3:44 PM Jens Axel Søgaard wrote:
 SandBox Text Files
 This is a simple example of writing to and reading from a text file:
\begin{axiom}
f1: TextFile := open("/tmp/test", "output")
writeLine!(f1, "abcdefg")
close! f1
f2: TextFile := open("/tmp/test", "...
 SandBox Trace Analysed
 This page is work in progress, to be continued ...
 SandBox Trace EXPR and FRAC
 Trace may affect correct output of '1::EXPR INT' . This page should be read in conjunction with IssueTracker #283.
To suppress the infinite loop being displayed till overflow, add ')math' to the ...
 SandBox Trace in Windows
 I want to point out a bug in Windows version: where the Linux version below refers to
'Integer.=,64' the Windows version (shown next) shows 'Integer.random,63'.
 SandBox Trace with )math
 The addition of ')math' below stopped the infinite loop from occurring!
For the version without ')math', see [SandBox Trace in Windows] ::
 SandBox Zero
 We are trying to create a Zero
 SandBox kaveh
 \begin{axiom}
 SandBox polymake
 Here we demonstrate the PolyMake wrappers 'Polytope' and 'SimplicialComplex'.
 SandBoxTDT
 SandBox.GuessingSequence
 This page makes test uses of the guessing package by Martin Rubey. Feel
free to add new sequences or change the sequences to ones you like to try.
 SandBox/interp/ifunsel.boot
 The purpose of this page is to begin the process of documenting
the function selection process used by the FriCAS interpreter.
 SandBox11
 SandBox12_Test_Indet_and_Complex
 \begin{axiom}
2^0.5
1^0.5
sqrt(2) :: Float
sqrt(1) :: Complex Float
0^0
0.0^0.0
\end{axiom}
 SandBox2
 \begin{axiom}
)set output tex on
)set output algebra off
)set output mathml off
\end{axiom}
 SandBox3
 quad Bezier/line Intersection
 SandBox42
 \begin{axiom}
 SandBox9
 looking for completion of pendulum (constrained motion)
 SandBoxA000230
 \begin{axiom}
a: (Integer) > Integer
a(n) ==
i:=2
repeat
j:=nextPrime(i)
if ji = 2*n or n = 0 then
return i
i:=j
 SandBoxAbelianDuck
 During the Aldor/Axiom Workshop 2008 at RISC in one of his
presentations Stephen Watt began with::
 SandBoxAbs
 This is how FriCAS currently differentiates the absolute value function.
\begin{axiom}
differentiate(abs(x),x)
\end{axiom}
 SandBoxActiveAxiomDevelopers
 The Core Axiom developers in alphabetical order
 SandBoxAdjacencyMatrix
 *On Tue, Jun 17, 2008 at 10:07 AM Franz Lehner wrote:*
 SandBoxAldorDomainList
 This is an experimental page to implement cartesian product over an indexed set of domains in Aldor. The constructor Product constructs a domain which is the cartesian product of $f(i)$ for $i \in I$.
 SandBoxAldorInductiveTypes
 This implementation of an Inductive (recursive) data type involves
the categorical coproduct construction. See: LimitsAndColimits
\begin{axiom}
)library colimits
\end{axiom}
 SandBoxAldorTesting
 Based on the Aldor presentation by Stephen Watt:
http://www.aldor.org/docs/reports/ukqcd2000/intro1ukqcd00.pdf
and
http://www.aldor.org/docs/HTML/chap2.html
 SandBoxAldorTuples
 **On Date: Jun 26, 2007 8:19 AM Ralf Hemmecke wrote:((
 SandBoxAlgebraOfObservers2
 Obs(2) is a 4 dimensional Frobenius Algebra
 SandBoxAlgfunc
 Add conjugate
\begin{spad}
)abbrev category ACF AlgebraicallyClosedField
++ Author: Manuel Bronstein
++ Date Created: 22 Mar 1988
++ Date Last Updated: 27 November 1991
++ Description:
++ Model for ...
 SandBoxAxiomAndThe30YearHorizon
 Intent
 SandBoxBen2
 \begin{axiom}
4^(4^(3)+2)
\end{axiom}
 SandBoxBiQuaternions
 The Biquaternion calculus support function collection
 SandBoxBiQuaternions2
 Biquaternion Calculus Domain
 SandBoxBiQuaternions3
 Biquaternion Calculus Domain
 SandBoxBootStrap
 On Monday, October 16, 2006 7:30 PM Gabriel Dos Reis wrote:
 SandBoxBootStrap2
 \begin{spad}
)lisp (setq $bootStrapMode t)
)abbrev domain T T
 SandBoxBugFactor
 \begin{axiom}
t:=(4*x^3+2*y^2+1)*(12*x^5x^3*y+12)*(x+1)*(y^2+3)*(x^21)
factor t
\end{axiom}
 SandBoxBugGroebnerFactorize
 \begin{axiom}
ideq:LIST(POLY(FRAC(SMP(INT,OVAR([mp,mq,mr,γqp,γrp,γrq]))))) := [ _
(γrp^2*mr^2*mp*%x5+(γrq*γrp*γqp*mr*mq*mp*%x4+γrp^2*mr*mp*%x1))*%x8+((γrq*γrp*γqp*mr*mq*mp*%x5+(γqp^2*mq^2*mp*%x4+γ...
 SandBoxBugLaurentSeries
 \begin{axiom}
Z==>Integer; Q==>Fraction Z
z: Symbol := 'z; P==>UnivariatePolynomial(z,Q)
t:P := monomial(1,1)
p:P := (1t)*(1t^2)*(1t^3)
L==>UnivariateLaurentSeries(Q,z,0)
R ==> Record(k: Z, c: Q)
...
 SandBoxCLWEB
 \documentclass{article}
\usepackage{axiom}
\usepackage{geometry}
\usepackage{hyperref}
\geometry{verbose,letterpaper,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\usepackage{graphics}
\...
 SandBoxCS224
 \begin{axiom}
artanh:(Float) > Float
artanh(x) ==
y := x
k := 1
for k in 1..precision() repeat
z := x^(2*k+1) / (2 * k + 1)
y := z + y
return y
 SandBoxCachedFunction
 A generic way to define function that cache their values.
 SandBoxCartesianTensor
 Try to make it a little faster.
\begin{spad}
)abbrev domain CARTEN CartesianTensor
++ Author: Stephen M. Watt
++ Date Created: December 1986
++ Date Last Updated: May 15, 1991
++ Basic Operations:
++ ...
 SandBoxCategories
 This page was renamed to AnonymousCategories. You can delete this one if no longer needed.
 SandBoxCategoryTerms
 The aim of this page is to relate mathematical terms, such as those from category theory or type theory, to the way these terms are used in SPAD.
So a reader who has some acquaintance with a given ...
 SandBoxChoose
 \begin{equation}
\left( {\begin{array}{*{20}c}
S \\
k \\
\end{array}} \right)
\end{equation}
 SandBoxCliffordAlgebra
 We want to test some properties of the CliffordAlgebra domain implemented in FriCAS
 SandBoxClosedLinearOperator
 Introduction
 SandBoxCoMultiplication
 \begin{axiom}
n:=2
T:=CartesianTensor(1,n,FRAC POLY INT)
Y:T := unravel concat concat
[[[script(y,[[i,j],[k]])
for i in 1..n]
for j in 1..n]
for k in 1..n]
A:T := unravel(concat ...
 SandBoxCollect
 **Collect terms in x with given factor k.**
\begin{spad}
)abbrev package COLL Collect
I==>Integer
E==>Expression I
K==>Kernel E
Q==>PolynomialCategoryQuotientFunctions( _
IndexedExponents K, K, ...
 SandBoxCombfunc
 Add conjugate
\begin{axiom}
)library BOP BOP1
\end{axiom}
\begin{spad}
)abbrev category COMBOPC CombinatorialOpsCategory
++ Category for summations and products
++ Author: Manuel Bronstein
++ Date ...
 SandBoxCombfuncDiscussion
 Nicer display from binomials:
\begin{axiom}
)set output tex off
)set output algebra on
binomial(n,m)
opbinom := operator(operator 'binomial)$CombinatorialFunction(INT,EXPR INT)
setProperty(...
 SandBoxCombinat
 This page demonstrates one of the very early attempts to do something similar to MuPAD combinat's "decomposable objects" in Aldor/Axiom. Meanwhile, we adopted the point of view of species, as shown ...
 SandBoxCommutativeCategory
 Note: In FriCAS the 'commutative("*")' attribute is replaced by
'CommutativeStar' catogory. The disscussion below is kept as
it raises some issues that are still relevant.
 SandBoxComplementsdalgebrelineaire
 \documentclass{article}
%
% linalg.ltx
% Francois Maltey  janvier 2008
%
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage[french]{babel}
\usepackage{xspace}
\...
 SandBoxComplexFunctionTest
 \begin{axiom}
Gamma(0.5+0.5*%i)
\end{axiom}
 SandBoxComplexManifold
 Complex domain constructor done differently.
 SandBoxConditionalFunctions
 **Conditional for inner functions in a package.**
 SandBoxDFORM
 \begin{spad}
)abbrev package DFORM DifferentialForms
++ Author: Kurt Pagani
++ Date Created: October 2014
++ Revised: Sat Sep 17 17:10:00 CET 2016
++ License: FriCAS/BSD
++ Description: Basic ...
 SandBoxDFT
 \begin{spad}
)abbrev package TYPEPKG TypePackage
TypePackage (T : Type) : Exports == Implementation where
Exports == with
typeof : T > Type
Implementation == add
typeof(x:T) == T
 SandBoxDGE
 Something is going wrong with the formatting of the output.
\begin{axiom}
s:=integrate(sqrt(a*t^2+b*t+c),t);
#s
s.1
s.2
\end{axiom}
It is not being wrapped as expected.
 SandBoxDOPT
 \begin{spad}
)abbrev package DOPT DiscreteOptimalTransport
++ Author: Kurt Pagani
++ Date Created: Thu Jun 21 01:42:00 CEST 2018
++ License: BSD
++ References:
++ Description:
++
...
 SandBoxDefineInteger
 Why are the domains PositiveInteger and NonNegativeInteger defined as
[SubDomain]s of Integer insteadof the other way around? Here is a
(still somewhat imperfect) example of one way of defining ...
 SandBoxDemoXFreeAlgebra
 \begin{axiom}
 Category: XFreeAlgebra(vl : OrderedSet, R : Ring) : Category
 ++ Author: Michel Petitot petitot@lifl.fr
 ++ Date Created: 91
 should be enough to impl TensorAlgebra(RModule), ...
 SandBoxDifferentialEquations
 Differential Equations
 SandBoxDifferentialGeometry
 Subject: DeRhamComplex
 SandBoxDifferentialPolynomial
 Description 
DifferentialSparseMultivariatePolynomial implements an ordinary differential polynomial ring by combining a domain belonging to the category DifferentialVariableCategory with the ...
 SandBoxDiracDelta
 Distributions
 SandBoxDiscussion
 Time to clean out the SandBox!
 SandBoxDomainConstructor
 \begin{axiom}
)set output tex off
)set output algebra on
\end{axiom}
 SandBoxDomainOfComputation
 An "assume" system based on equational reasoning.
 SandBoxDoublePowerSeries
 \begin{axiom}
a!
!:=operator '!
a(i,j)==(1+(j1)*i)/!(2*i+4*j)
a0:=matrix [[a(i,j)*k^i*l^j for i in 0..5] for j in 1..4]
a(i:INT,j:INT):FRAC INT == (1+(j1)*i)/Gamma(1+2*i+4*j)
aa:DMP([k,l], FRAC INT)...
 SandBoxDoyen
 PRECOMPILED BINARIES
 SandBoxEcfact
 Subject: aldor/axiom interoperability
 SandBoxElementaryFunctionStructurePackage
 \begin{axiom}
)lib FSPECX
\end{axiom}
 SandBoxElemntry
 Add conjugate
\begin{spad}
)abbrev package EF ElementaryFunction
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Date Last Updated: 10 April 1995
++ Keywords: elementary, function, logarithm, ...
 SandBoxEllipticCurves
 Elliptic curve computations mod N in projective coordinates
 SandBoxEqualizerInOpenAxiom
 In FriCAS 'Equalizer' (See: LimitsAndColimits) can be defined as a
'SubDomain' as follows, but this does not work in the most recent
version of OpenAxiom
 SandBoxEtc
 \begin{axiom}
integrate(x, x)
\end{axiom}
 SandBoxExpOfEnd
 Exponential of endomorphism with minimal polynomial
 SandBoxExpr
 Add conjugate
\begin{spad}
)abbrev domain EXPR Expression
++ Toplevel mathematical expressions
++ Author: Manuel Bronstein
++ Date Created: 19 July 1988
++ Date Last Updated: October 1993 (P.Gianni),...
 SandBoxExpression
 \begin{axiom}
fm:=FreeMonoid(OVAR [e1,e2])
EM:=Expression(fm)
x1:=e1::Symbol::EM
x2:=e2::Symbol::EM
p:=operator('p)
p(x1)
x1*x2
\end{axiom}
 SandBoxExpressionExpression
 \begin{axiom}
x:=coerce('x)$Expression(Expression Integer)
ground(x)
variables x
ex1:=sin(x)$Expression(Expression Integer)
kernels ex1
\end{axiom}
 SandBoxExpresssionComplex
 \begin{axiom}
msqrt:=operator('msqrt)
conj1:Ruleset(Integer,Complex Integer,Expression Complex Integer) := ruleset([ _
rule sqrt(1)*:a==msqrt(1)*a, _
rule sqrt(1)*:a==msqrt(1)*a _
]$List ...
 SandBoxF3lix
 \begin{axiom}
myExpr := (1e^x)^2  (e^(2*x) 2*e^x + 1)
\end{axiom}
 SandBoxFORMAN
 \begin{spad}
)abbrev package FORMAN FormalManipulations
++ Author: Kurt Pagani
++ Date Created: Tue Jun 19 19:09:16 CEST 2018
++ License: BSD
++ References:
++ Description:
++ Interactive Computer ...
 SandBoxFSpace
 Add conjugate
\begin{axiom}
)lib BOP,BOP1,FSPECF
\end{axiom}
 SandBoxFileUploadTest
 Here is a patch:
 SandBoxFisher
 How useful are the different CAS languages for implementing numerical routines? Prompted by a comparison of R and C for implementing Fisher's exact test for 2x2 tables (http://fluff.info/blog/arch/...
 SandBoxFloatSegment
 Split from #358
 SandBoxFormalProduct
 \begin{axiom}
f:=operator 'f
differentiate(f(x),x) = differentiate(f(y),y)
test %
integrate(f(x),x) = integrate(f(y),y)
test %
 y is replaced with x?
product(f(x),x) = product(f(y),y)
test %
...
 SandBoxFormann
 f calculates the number of 01 matrices with row sums 'A' and column sums 'B'.
 SandBoxFractionGCD
 \begin{axiom}
R := Integer
Q := Fraction R
gcd(8,4) = retract gcd(8::Q, 4::Q)
\end{axiom}
 SandBoxFreeProduct
 \begin{spad}
)abbrev domain FPROD FreeProduct
++ Description:
++ This domain implements the free product of monoids (groups)
++ It is the coproduct in the category of monoids (groups).
++ FreeProduct(...
 SandBoxFreeSum
 \begin{spad}
)abbrev domain FSUM FreeSum
++ Description:
++ This domain implements the free "product" of abelian monoids (groups)
++ It is the coproduct in the category of abelian monoids (groups).
++...
 SandBoxFriCAS
 This page is set to execute !\begin{axiom}... \end{axiom}
commands using FriCAS. See also FriCASIntegration.
\begin{axiom}
)version
\end{axiom}
 SandBoxFriCAS2
 Try FriCAS calculations here. For example, here is a simple FriCAS command::
 SandBoxFrobeniusAlgebra
 References
 SandBoxFunctionalSpecialFunction
 Changed derivative of abs(x) to
$$
\frac{\overline{x}}{2\ abs(x)}
$$
Added conjugate(x).
 SandBoxFunctor
 Gaby described how to obtain the type of a functor. A functor is an "unapplied" domain constructor. A domain constructor is an expression. So for example 'Complex Integer' is a domain constructor. '...
 SandBoxGenericFunction
 Generic Functions in FriCAS
 SandBoxGeom1
 \begin{spad}
)abbrev domain RSPACE RealSpace
++ Author: kfp
++ Date Created: Thu Nov 06 03:51:56 CET 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations:
++ Related ...
 SandBoxGlossary
 **Glossary**
 SandBoxGnuDraw
 \begin{axiom}
)clear completely
m(0,p) == 0
m(n,p) == m(n1,p) * p + 1
\end{axiom}
 SandBoxGnuPlotTex
 Gee, that's easy:
\begin{gnuplot}[terminal=pslatex,terminaloptions=color,scale=1.4]
plot [pi:pi] cos(x) lw 4,sin(x) lw 4
\end{gnuplot}
 SandBoxGradedTensor
 \begin{spad}
)abbrev domain GTEN GradedTensor
GradedTensor(n:NonNegativeInteger, m:NonNegativeInteger, R:CommutativeRing,dim:NonNegativeInteger): Join(GradedAlgebra(R, NonNegativeInteger), ...
 SandBoxGraphviz
 \begin{latex}
\digraph[scale=1.0]{SandBox5Graph1}{rankdir=LR; a>b>c; a>h; b>d; c>h; e>d e>f>g; f>h; g>h}
\end{latex}
 SandBoxGrassmannIsometry
 Isometry from Grassmann Multivectors
 SandBoxGroebnerBasis
 Adapted from "Ideals, Varieties, and Algorithms Third Edition, 2007":http://www.cs.amherst.edu/~dac/iva.html
 SandBoxHermitianIsomorphism3x
 A complex vector CCspace $V$ possesses many different hermitian isomorphisms
$h^\dagger=h \in iso(V,V^\dagger)$. In quantum mechanics a given operator
$\rho \in End(V)$ may be said to be $h$...
 SandBoxHermitianIsomorphisms
 A complex vector ℂspace $V$ possesses many different hermitian isomorphisms
$h^\dagger=h \in iso(V,V^\dagger)$. In quantum mechanics a given operator
$\rho \in End(V)$ may be said to be $h$...
 SandBoxHermitianIsomorphisms3
 A complex vector ℂspace $V$ possesses many different hermitian isomorphisms
$h^\dagger=h \in iso(V,V^\dagger)$. In quantum mechanics a given operator
$\rho \in End(V)$ may be said to be $h$...
 SandBoxHermitianIsomorphisms4
 EigenVectors and Diagonalization
 SandBoxHiddenOverloading
 This page discusses cases where instantiating of a parameter collapses two generically different functions.
 SandBoxHopfAlgebra
 **On Fri, May 15, 2009 at 03:23:33AM +0200, Franz Lehner wrote:**
 SandBoxHyperDocReplacement
 Here I demonstrate a possible way to start a replacement for HyperDoc.
 SandBoxHyperGeometric
 \begin{spad}
)abbrev package SUMMPACK SummPack
SummPack(): Exports == Implementation where
FPI ==> Fraction Polynomial Integer
X ==> Expression Integer
NNI ==> NonNegativeInteger
PI ==> ...
 SandBoxINFSUM
 \begin{spad}
)abbrev package UNIFY Unify
++ Author: Kurt Pagani
++ Date Created: Mon Feb 06 18:56:20 CET 2017
++ License: BSD
++ References:
++ Description:
++
Unify(X) : Exports == Implementation ...
 SandBoxIOutput
 Example for debugging boot code
 SandBoxInductiveType
 **On May 6, 2007 11:01 PM Gabriel Dos Reis wrote:**
 SandBoxInequation
 \begin{spad}
)abbrev domain NEQ Inequation
++ Author: Bill Page
++ Based on: Equation by Stephen M. Watt, enhancements by Johannes Grabmeier
++ Date Created: June 2008
++ Basic Operations: ~=
++ ...
 SandBoxJacobiDiag
 \begin{spad}
)abbrev package JACDIAG JacobiDiagonalisation
++ Author: Kurt Pagani
++ Date Created: Sat Jun 16 23:37:27 CEST 2018
++ License: BSD
++ References:
++ Jacobi, C.G.J. (1846). "Über ein ...
 SandBoxJacobiDiagFloat
 \begin{axiom}
PI ==> PositiveInteger
NN ==> NonNegativeInteger
IF ==> Float
VIF ==> Vector Float
MIF ==> Matrix Float
 SandBoxJacobiDiagIntervalFloat
 \begin{axiom}
RR ==> Expression Integer
CC ==> Expression Complex Integer
 SandBoxJacobianMatrix
 \begin{axiom}
)expose MCALCFN
jacobian([x^2+y,x*y],[x,y]::List Symbol)
\end{axiom}
 SandBoxJohn2
 This is a new SandBox test page. Since the page name begins with
SandBox, no email notice should be generated if this page is
changed.
 SandBoxKMG
 \begin{axiom}
 SandBoxKernel
 \begin{spad}
)abbrev category PARTSET PartitionedSet
++ Sets whose elements are grouped into equivalence classes by a mapping
++ Author: Bill Page
++ Date Created: 20 March 2015
++ Description:
++ ...
 SandBoxLeftFreeModule
 \begin{spad}
)abbrev domain LFREEMOD LeftFreeModule
LeftFreeModule(R: Ring, S: OrderedSet):
Join(LeftModule R, IndexedDirectProductCategory(R,S)) with
linearCombination: List Product(S,R) >...
 SandBoxLexicalScope
 j is referenced out of scope
\begin{axiom}
for i in 1..5 repeat
j:Integer := i^2
output(i::OutputForm)$OutputPackage
output(j::OutputForm)$OutputPackage
\end{axiom}
The code above ...
 SandBoxLimits
 \begin{axiom}
1/sqrt(x^2 + 1)  sqrt(x^2 + 1)/(x^2 + 1)
limit(1/sqrt(x^2 + 1),x=%i)
limit(sqrt(x^2 + 1)/(x^2 + 1),x=%i)
\end{axiom}
 SandBoxLinearOperator
 \begin{axiom}
)lib CARTEN MONAL PROP
\end{axiom}
 SandBoxLinearProgramming
 \begin{spad}
)abbrev package LINPROG LinearProgramming
++ Author: Kurt Pagani
++ Date Created: Sun Jun 03 02:05:55 CEST 2018
++ License: BSD
++ References:
++ Description:
++ TODO: description, more ...
 SandBoxListLength
 \begin{spad}
)abbrev category MYLIST MyListPackage
MyListPackage(T: Type): with
length2: List T > Integer
== add
len(count: Integer, l: List T): Integer ==
empty? l => count
...
 SandBoxLiteral
 Literals and Symbols in Axiom
 SandBoxLorentzTransformation
 Lorentz transformations.
 SandBoxMIWHall
 Many Interacting Worlds  Hall, et al.
\begin{axiom}
P:=operator('P);
Q:=operator('Q);
g:=operator('g);
U:=operator('U);
r:=operator('r);
ℏ:=h;
\end{axiom}
 SandBoxMLE
 I have recently been experimenting with using Axiom for maximum likelihood estimation,
borrowing code (with permission) from "Compact Numerical Methods" by Nash.
 SandBoxMatrixExample
 Matrix Example
 SandBoxMaxSingleInteger
 The maximum of FriCASIssues is not really maximal.
 SandBoxMaxima
 Testing the Maxima interface
 SandBoxMaxima2
 Minimal Maxima
 SandBoxMaxima3
 Computing Christoffel symbols in Maxima
\begin{maxima}
load(ctensor);
depends([a,b],[t,r]);
ct_coords:[t,r,O,P];
lg:ident(4);
lg[1,1]:exp(2*a);
lg[2,2]:exp(2*b);
lg[3,3]:r^2;
lg[4,4]:r^2*sin(O)^2;
...
 SandBoxMaybe
 SandBoxMonad
 This may be a better definition of Monad in Axiom.
 SandBoxMonads
 **Bertfried Fauser** wrote:
 SandBoxMorphism
 Using OpenAxiom
\begin{axiom}
)version
\end{axiom}
 SandBoxMultivariatePolynomialDivision
 A tutorial written by Angelos Mantzaflaris (webpage:
http://users.uoa.gr/~amantzaf/proje.html )
 SandBoxMyReduce
 \begin{spad}
)abbrev package MYRED MyReduce
MyReduce(S:Type): with
myred: ((S,S)>S,List S) > S
== add
 Waldek's local helper function
((f:(S,S)>S) = (g:(S,S)>S)):Boolean ==
...
 SandBoxMyReduceInFriCAS
 \begin{axiom}
)version
\end{axiom}
\begin{spad}
)abbrev package MYRED MyReduce
MyReduce(S:Type): with
myred: ((S,S)>S,List S) > S
== add
 Waldek's local helper function
((f:(S,S)>S) = ...
 SandBoxMyReduceInOpenAxiom
 \begin{axiom}
)version
\end{axiom}
\begin{spad}
)abbrev package MYRED MyReduce
MyReduce(S:Type): with
myred: ((S,S)>S,List S) > S
== add
 Waldek's local helper function
((f:(S,S)>S) = ...
 SandBoxNewAutodocPamphlet
 \documentclass{article}
\usepackage{axiom}
\usepackage{hyperref}
\begin{document}
\title{NewAutodoc}
\author{William Sit}
\date{Original by Tim Daly\\ New Documentation August 13, 2006\\ Last updated ...
 SandBoxNewtonsMethod
 The following shows Newton's method for numerically solving f(x)=0. It is also shows examples of calling Axiom expressions and Spad functions from Lisp.
 SandBoxNijenhuis
 \documentclass[12pt,twoside]{article} %Marcel Dekker 2004
\usepackage{latexsym,amssymb,amsmath,amsfonts,amscd,amsthm,amstext}
\usepackage[all]{xy} %epsfig,graphics
%\setcounter{page}{1}
\...
 SandBoxNonAssociativeAlgebra
 Ref: http://arxiv.org/abs/0711.3220
 SandBoxNonZeroInteger
 Algebra
 SandBoxNonZeroIntegerInOpenAxiom
 Algebra
 SandBoxObserverAsIdempotent
 Obs(3) is a 9 dimensional Frobenius Algrebra
\begin{axiom}
)set output abbreviate on
)set message type off
\end{axiom}
 SandBoxObserverAsIdempotent2
 Obs(2) is a 4 dimensional Frobenius Algebra
 SandBoxObserverAsIdempotent4
 Obs(4) is a 16 dimensional Frobenius Algrebra
\begin{axiom}
)set output abbreviate on
V := OrderedVariableList [p,q,r,s]
M := FreeMonoid V
gens:List M := enumerate()$V
divisible := Record(lm: M,rm: M)...
 SandBoxOldFrontPage
 Welcome to "Axiom":AboutAxiom!
 SandBoxOp
 Conjugate opertors
\begin{spad}
)abbrev domain BOP BasicOperator
++ Basic system operators
++ Author: Manuel Bronstein
++ Date Created: 22 March 1988
++ Date Last Updated: 11 October 1993
++ ...
 SandBoxOpenAxiom
 What version?
\begin{axiom}
)version
\end{axiom}
 SandBoxOperatorsAndSqrt
 This fails unexpectedly:
\begin{axiom}
f:=operator 'f
f(sin(1))
f(sqrt(1))
\end{axiom}
 SandBoxOutputPackage
 \begin{axiom}
)set output algebra on
\end{axiom}
 SandBoxOverloading
 This fails in SPAD:
\begin{spad}
)abbrev domain AAA Aaa
Aaa: with
bar: Integer > Integer
== add
foo(x:Float):Integer == 1
foo(x:Integer):Integer == 2
a:Integer == 1
a:Float == ...
 SandBoxPQTY
 \begin{spad}
++++++++++++++++++++++++
++ $Id: 01AUG2015$ +++
++++++++++++++++++++++++
)abbrev domain RIA Q_INTERVAL
++ Author: Kurt Pagani
++ Date Created: 2012
++ License: BSD (same as Axiom)
++ ==...
 SandBoxPamphlet
 \documentclass{article}
\usepackage{axiom}
\usepackage[final,pdfstartview=FitH]{hyperref}
\begin{document}
\title{\$SPAD/src/algebra dhmatrix.spad}
\author{Richard Paul and Timothy Daly}
\maketitle
\...
 SandBoxPartialFraction
 Francois Maltey wrote:
 SandBoxPartiallyOrderedSet
 *On Tue, Aug 5, 2008 at 4:20 PM, Gabriel Dos Reis wrote:*
 SandBoxPauliAlgebra
 The Pauli Algebra Cl(3) Is Frobenius In Many Ways
 SandBoxPfaffian
 \documentclass{article}
\usepackage{amsfonts}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input pfaffian}
\author{Timothy Daly, Gunter Rote, Martin Rubey}
\maketitle
\begin{abstract}
In ...
 SandBoxPhysicalUnitSystem
 \begin{spad}
)abbrev category PUNIT PhysicalUnit
PhysicalUnit() : Category == with
"*" : (%,%) > %
"/" : (%,%) > %
"^" : (%,Integer) > %
"=" : (%,%) > Boolean
dim : % ...
 SandBoxPileNotation
 Example of wrong indentation.
 SandBoxPlotly
 \begin{spad}
)abbrev package PLOTLY Plotly
++ Author: Kurt Pagani
++ Date Created: Sat Jul 14 01:52:32 CEST 2018
++ License: BSD
++ References:
++ Description:
++
Plotly(R) : Exports == ...
 SandBoxPoirier
 Hall
 SandBoxPolynomialQuiz
 \begin{axiom}
)clear all
variables (2*x+1/x)$DMP([x], EXPR INT)
variables (2*y+1/y)$DMP([y], INT)
a:=(2*x+1/x)$DMP([x], EXPR INT); variables a
b:=(2*y+1/y)$DMP([y], INT); variables b
x:DMP([x], EXPR ...
 SandBoxPolynomialQuizExplained
 Coming Soon! (Or you can provide them!) Please insert comments after each problem.
Original quizzes on SandBoxPolynomials#quizzes
 SandBoxProduct
 \begin{spad}
)abbrev domain PRD2 Product2
rep(x) ==> x @ % pretend Rep
per(x) ==> x @ Rep pretend %
Product2(X:SetCategory,Y:SetCategory): with
construct: (X,Y) > %
coerce: % > OutputForm
...
 SandBoxProp
 \begin{spad}
)abbrev domain INEQTY InEquality
++ Author: kfp
++ Date Created: Sun Oct 26 02:21:23 CEST 2014
++ License: BSD (same as Axiom)
++ Date Last Updated:
++ Basic Operations:
++ Related ...
 SandBoxREN
 \begin{axiom}
As := matrix([ [3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])
A := subMatrix(As, 2,4,2,4)
ob := orthonormalBasis(A)
P : Matrix(Expression Integer) := new(3,3,0)
setsubMatrix!(P,1,1,ob.3) ...
 SandBoxRalf
 \begin{axiom}
E ==> Expression Integer
OF ==> OutputForm
macro R == Record(OUTPUTFORM:OF,_
SEXPRESSION: SExpression,_
SEXPRESSION: SExpression,_
...
 SandBoxRealSpace
 \begin{spad}
)abbrev domain RR Real
++ ===================================
++ THE REAL LINE MODELLED AS EXPR(INT)
++ ===================================
++ Modified version of Expression R
++ ...
 SandBoxReduce
 Try Reduce calculations here. For example::
 SandBoxRelativeVelocity
 \documentclass{slides}
\usepackage{latexsym,amssymb,amsmath,amsfonts,amscd,amsthm,amstext}
\newcommand{\bk}{\Bbbk}\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}\newcommand{\R}{\mathbb{R}}
\...
 SandBoxRemember
 On Tue, Oct 27, 2009 at 4:19 PM, Franz Lehner wrote::
 SandBoxRepresentationTheory
 From: input/repa6.input
\begin{axiom}
 SandBoxRibbonCategory
 Refs:
 SandBoxRootOfUnity
 \begin{spad}
)abbrev domain ROU RootOfUnity
++ Author: Kurt Pagani
++ Date Created: Fri Jun 01 17:24:19 CEST 2018
++ License: BSD
++ References:
++ https://en.wikipedia.org/wiki/Root_of_unity
++ ...
 SandBoxRotationMatrix
 This says don't use LaTeX output.
\begin{axiom}
)set output tex off
)set output algebra on
\end{axiom}
 SandBoxSEXPM
 \begin{spad}
)abbrev package UNIFY Unify
++ Author: Kurt Pagani
++ Date Created: Mon Feb 06 18:56:20 CET 2017
++ License: BSD
++ References:
++ Description:
++
Unify(X) : Exports == Implementation ...
 SandBoxSPADParser
 Parsing FriCAS/Axiom code
 SandBoxSSPM
 \begin{spad}
)abbrev package SSPM SuperSimplePatternMatcher
++ Author: Kurt Pagani
++ Date Created: Mon Feb 06 18:56:20 CET 2017
++ License: BSD
++ References:
++ www.cs.northwestern.edu/academics/...
 SandBoxSTRING2SPAD
 Waldek Hebisch helped me to write the following Lisp code to call the SPAD compiler from the FriCAS/AXIOM interpreter.
 SandBoxSage
 This is a test of "Sage":http://modular.math.washington.edu/sage
in MathAction.
 SandBoxSageAxiomInterface
 \documentclass{article}
\def\code#1{{\tt #1}}
\def\url#1{{\it #1}}
\def\sage#1{{\bf #1}}
\def\note#1{#1}
\usepackage{axiom}
 SandBoxSagePamphlet
 \documentclass{article}
 SandBoxScriptedSymbol
 \begin{axiom}
L ==> List OutputForm;
rsl: List L := [[x1::OutputForm],[x2::OutputForm],[],[]];
sl: L := [commaSeparate(z)$OutputForm for z in rsl];
of: OutputForm := scripts(v::OutputForm, sl)$...
 SandBoxSignum
 Ref.
 SandBoxSimple
 SandBoxSinCosRules
 Sin and Cos Rules
 SandBoxSpecies
 Test species in axiom/FriCAS.
 SandBoxSpeciesAldor
 Test species in axiom/FriCAS.
 SandBoxSpeciesAldorSage
 Test species in axiom/FriCAS and compare with the implementation in
"SageCombinat":http://sage.math.washington.edu/home/mhansen/CombinatorialSpeciesDemo.html
 SandBoxSpeciesBayreuth
 \begin{axiom}
)cd ~/combinat/src
)re ../lib/combinat.input
\end{axiom}
 SandBoxSpeciesBayreuth2
 Test the different Series defined for Species
 SandBoxSpeciesCategory
 \begin{spad}
)abb category SPECCAT SpeciesCategory
SpeciesCategory(L: SetCategory): Category == SetCategory with
structures: Set L > List %
name: () > String
 SandBoxSqrt
 Square root
\begin{axiom}
ksqrt:=kernels(sqrt(x))(1)
name(ksqrt)
argument(ksqrt)
\end{axiom}
 SandBoxSum
 The Sum domain constructor is intended to be the
"Categorical Dual":http://en.wikipedia.org/wiki/Dual_(category_theory)
of the "Product":/axiomtest1/src/algebra/ProductSpad
domain constructor
\...
 SandBoxSurfaceComplex
 \begin{axiom}
)abbrev domain CMAP CellMap
++
CellMap(R,n) : Exports == Implementation where
 SandBoxSymPy
 Running SymPy in a SageBlock
 SandBoxSymbol
 Allow adding additional sub/superscripts to symbols.
\begin{spad}
)abbrev domain SYMBOL Symbol
++ Author: Stephen Watt
++ Date Created: 1986
++ Date Last Updated: 7 Mar 1991, 29 Apr. 1994 (FDLL)
++ ...
 SandBoxSymbolic
 Symbolic Integers
 SandBoxSymbolicExpressions
 Unevaluated mathematical expressions
 SandBoxSymbolicInverseTrig
 How does FriCAS compute inverse trigonometric functions symbolically?
 SandBoxSymbolicSumsAndProducts
 Symbolic evaluation of sums and products
 SandBoxSymbolicTrig
 \begin{axiom}
)lib AXIOM SINF SYMCAT SYMB AND SEXPR
)lib ALTAX ALTAX ALT
)lib ASSAX ASSAX ASSAX+ ASSAXS ASS+ ASSS
)lib CANAX CANAX CAN
)lib COMAX COMAX COMAX+ COM+ COMAX1 COMS COMAX2 COMAX3
)lib ...
 SandBoxSyntax
 \documentclass{article}
\usepackage{axiom}
 SandBoxTables
 StructuredText supports tables with a simpler markup than HTML
using this markup::
 SandBoxTensorAlgebra
 \begin{spad}
)lisp (setq $inclAssertions nil)
 https://en.wikipedia.org/wiki/Tensor_algebra
 SandBoxTensorAlgebra2
 \begin{spad}
)abbrev domain TENSALG TensorAlgebra
TensorAlgebra(M,R,B) : Exports == Implementation where
 SandBoxTensorAlgebra3
 \begin{spad}
)abbrev domain TENSALG TensorAlgebra
++ Author: Kurt Pagani
++ Date Created: Thu Jan 30 23:04:02 CET 2020
++ License: BSD
++ References:
++ https://en.wikipedia.org/wiki/
...
 SandBoxTensorPower
 \begin{axiom}
n:=5000  whatever PI
 SandBoxTensorProduct
 **Date: Fri, 15 May 2009 21:03:00 +0200 Franz Lehner wrote:**
 SandBoxTensorProduct3
 \begin{axiom}
R ==> EXPR INT
 SandBoxTensorProductPolynomial
 http://en.wikipedia.org/wiki/Tensor_product
 SandBoxTest1
 \begin{axiom}
1 + 2
\end{axiom}
 SandBoxTestTexOutput
 \begin{axiom}
integrate(cos(sqrt y)^3*y,y)
integrate(cos(sqrt y)*y,y)/3::OutputForm+(integrate(cos(sqrt z)*z,z)/3)::OutputForm
(z+1)^16
f:=operator 'f
differentiate(f(1/sin(x)),x)
integrate(1/sqrt(x^...
 SandBoxThisFunctor
 In a domain and in categories referenced in a domain the notation %
represents "this domain" (or self in some programming languages). So
we commonly write for example::
 SandBoxThisFunctorAldor
 In a domain and in categories referenced in a domain the notation %
represents "this domain" (or self in some programming languages). So
we commonly write for example::
 SandBoxThisFunctorFriCAS
 In a domain and in categories referenced in a domain the notation %
represents "this domain" (or self in some programming languages). So
we commonly write for example::
 SandBoxTraceAffectsResults
 \begin{axiom}
integrate(sin(sqrt(x*x)),x)
D(%,x)
%sin(sqrt(x*x))
normalize %
)trace INTEF )math
integrate(sin(sqrt(x*x)),x)
D(%,x)
%sin(sqrt(x*x))
normalize %
)trace INTEF )off
integrate(sin(sqrt(x*...
 SandBoxTroubleWithOperators
 Operators are not immutable. Operator properties are dynamic and among other things affect equality!
\begin{axiom}
f:=operator 'f
g:=operator 'f
test(f=g)
test(kernels f(x) = kernels g(x))
test(f(x)=...
 SandBoxTryOutSPAD
 Everyone is allowed to try out SPAD by editing this page.
Don't be affraid to destroy anything. The purpose of this page
is exactly to give a place for everyone to experiment.
 SandBoxUnify
 \begin{axiom}
)abbrev package EQREASON EquationalReasoning
++ Author: Kurt Pagani
++ Date Created: Mon Mar 21 17:10:18 CET 2016
++ License: BSD
++ References:
++ Description:
++
EquationalReasoning(R)...
 SandBoxUnitsAndDimensions
 See also the page named to [Units and Dimensions].
 SandBoxUnparse
 *On Tue, Sep 16, 2008 at 11:11 PM Waldek Hebisch wrote:*
 SandBoxVeryLongLaTeX
 Examples of FriCAS output that causes problems for the LineBreaker program:
 SandBoxWirtinger
 Wirtinger Derivatives or Wirtinger Calculus (also called $\mathbb{CR}$Calculus)
 SandBoxXHashTable
 A hash table implementation via double hashing
 SandBoxXvfbRun
 \begin{axiom}
viewport1 := draw(sin x, x=%pi..%pi)
write(viewport1, "SandBoxXvfbRun1", "postscript")
\end{axiom}
 SandBoxZtransform
 Alasdair McAndrew writes::
 SandBox[Polynomial Sequences: Matrix]
 \begin{axiom}
 Functions to facilitate Transforming Polynomial Generating Functions into
 Coefficient arrays. With some examples
 These are coefficient arrays (A) and can be rendered into ...
 SandBox_pleblond
 \begin{axiom}
solve(p=n*log(n),n)
\end{axiom}
 SandBox{{7*7}}
 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 ...
 Sandbox Errors
 This is collection of popular errors made when entering FriCAS expressions.
 Sandbox Variables Evaluation
 ** Thoughts on Variables Evaluation **
 SandboxDelay
 delay vs. generate  yield
 SandboxDemoFreeMonoid/Module
 \begin{axiom}
 FreeMonoid(S: SetCategory)
 S can be almost anything
 SandboxErik
 How do I find the nroot of a number?
 SandboxExportType
 \begin{spad}
)abbrev package TEST Test
Test(x: Ring): with
baseRing: () > Ring
== add
baseRing() == x
\end{spad}
\begin{axiom}
 SandboxFactoringNoncommutativePolynomials
 Konrad Schrempf wrote::
 SandboxFreeAndLocal
 \begin{spad}
)abbrev domain AAA Aaa
Aaa: with
foo: Integer > Integer
== add
z: Integer := 0
zzz: () > Integer
foo(n: Integer): Integer ==
z := 1
n + 2*zzz()
...
 SandboxIsomorphic
 This page deals with specification of isomorphic domains.
 SandboxNullaryVsConstant
 \begin{spad}
)abbrev domain FOO Foo
Foo(): Exports == Implementation where
Exports ==> with
myrandom: Integer
 myrandom: () > Integer
Implementation ==> add
myrandom == 42
 ...
 SandboxOutputBug
 \begin{axiom}
OF ==> OutputForm
R ==> Record(OUTPUTFORM:OF, SEXPRESSION: SExpression, TEXFORMAT:TexFormat)
e x ==> (print(([x::OF, (x::OF) pretend SExpression, x::OF::TexFormat]$R)::OF);x)
)set ...
 SandboxOutputTest
 \begin{axiom}
)version
OF==>OutputForm
%pi
message("%pi")$OF
message(" %pi")$OF
message("%pi ")$OF
"%pi"
%pii
message("%pii")$OF
"%pii"
message("\")$OF
\end{axiom}
 SandboxPamphletStyle
 This page describes how pamphlets should look like.
 SandboxPermutations
 \begin{axiom}
PI ==> PositiveInteger
LPS ==> List Permutation SetCategory
PS ==> Permutation SetCategory
LPSET ==> List Permutation Set Integer
PSET ==> Permutation Set Integer
lookup(lps2:LPS,i:PI): ...
 SandboxPlex
 \begin{axiom}
OF ==> OutputForm
 e x ==> x::OF pretend SExpression
e x ==> x
e integral(x^x, x)
e integral(x^x, x=a..b)
e sum(x^x, x)
e sum(x^x, x=a..b)
e product(x^x, x)
e product(x^x, x=a..b)
e ...
 SandboxSomos2Eta
 Translation from Somos notation to etafunction notation and back
 SandboxTypeDefinitions
 There seems to be different understandings of Type, domain, category, Category, etc. around.
Here is an attempt to collect all these different opinions in order to make discussion about them
clearer.
 SandboxWorkShopRISC2008
 Aldor & Axiom Workshop 2008
part of
"RISC Summer 2008":http://www.risc.unilinz.ac.at/about/conferences/summer2008/
 ScratchPad
 Scratchpad was a large, general purpose computer algebra system
developed at IBM. Developement was started around 1965 by
J. Griesmer with creation of Lisp system. Later,
Richard Jenks joined ...
 ScreenShots
 A short demonstration
 SequenceIteration
 Sequence Iteration
 SeriesSolve
 This package is included in FriCAS. Compared to previous series solver
it handles quite arbitrary functional equations, but uses a questionable algorithm.
It certainly should be replaced by ...
 SimpleSpadExamples
 This page is a hub to link to various simple examples. Currently: PascalTriangle,
SequenceIteration, SymbolicDifferentiation, HeapSort1.
 Simpson's method
 This routine provides Simpson's method for numerical integration. Although FriCAS already provides a Simpson's method, this version has a syntax that will be intuitive to anyone who has used the ...
 SiteIndex
 A glossary generated from the pages on this site.
 Snake Relation
 Nondegeneracy of the pairing
 Solving Differential Equations
 Test 1: solve a simple nonlinear homogeneous differential equation
 Source Repository
  [FriCAS Sources]
 SourceForge
 RemoteUrl: http://sourceforge.net
 SpadFileConvention
 Conventions for writing .spad files after noweb has been dropped
 SparseMultivariateTaylorSeries
 How to work with multivariate Taylor series
 SpeedOfUserCode
 Here we show FriCAS version of benchmark presented at
http://cosmosimple.blogspot.de/2013/12/somebenchmarksofmathematicavs.html
 Sqrt3Demo
 Some demo involving the algebraic number $\sqrt{3}$.
 Status Page
 Adapted from ZopeWikiStats
 StructuredTables
 \begin{axiom}
P:= [[1,0,0],[0,1,0],[0,0,0]]
Q:= [[0,0,0],[0,0,0],[0,0,1]]
CT := CARTEN(i0 := 1,3,Integer)
epsilon:CT:=leviCivitaSymbol()
permi1:= matrix[[e0,0,0],[0,e0,0],[0,0,e0]]
permi2:= ...
 StructuredText
 Structured Text is a simple plain text markup format used by Zope
and ZWiki. It's similar in effect to !WikiWikiWebMarkup but is more
intuitive and standardized. The idea is to have text that is
...
 SubDomain
 PositiveInteger and NonNegativeInteger are subdomains of Integer.
 SymbolInteger
 \begin{spad}
)abbrev package SYMINT SymbolInteger
++ Date Created: Fri Sep 02 01:11:15 CEST 2022
++ Description: _[]d+ ; symbols as integers
++
SymbolInteger(T:Type) : Exports == Implementation ...
 Symbolic Integration
 Errors in symbolic integration
 SymbolicAssociativeExpressions
 \begin{axiom}
)lib AXIOM SINF SYMCAT SYMB AND SEXPR
)lib ALTAX ALTAX ALT
)lib ASSAX ASSAX ASSAX+ ASSAXS ASS+ ASSS
)lib CANAX CANAX CAN
)lib COMAX COMAX COMAX+ COM+ COMAX1 COMS COMAX2 COMAX3
)lib ...
 SymbolicDifferentiation
 Symbolic Differentiation Tutorial
 SymbolicItoCalculus
 itovsn3 (author: Wilfrid S. Kendall)
 SymbolicNonAssociativeExpressions
 \begin{axiom}
)lib AXIOM SINF SYMCAT SYMB AND SEXPR
)lib ALTAX ALTAX ALT
)lib ASSAX ASSAX ASSAX+ ASSAXS ASS+ ASSS
)lib CANAX CANAX CAN
)lib COMAX COMAX COMAX+ COM+ COMAX1 COMS COMAX2 COMAX3
)lib ...
 SymbolsAndVariables
 Interpreter
 series
 Currently FriCAS can not evaluate
 simplify exponents
 How to simplify exponents
 simplifying Expressions
 Simplification of Expressions
 statistical functions
 Anonymous shared the following library of statistical functions with us. Many thanks! (He added, that they are not checked...)
 string conversions
 Here is a package that converts strings to numbers. It is only to be considered as a starting point. There is no error checking, it is probably slow and certainly buggy...
T  TaylorSeries
 Note 04Apr2021: This page is abandoned, it's new home is
https://fricas.github.io/fricasnotebooks/FriCASTaylorSeries.html
 TeX
 and LaTeX are mathematical typesetting programs.
 TeXmacs
 "GNU TeXmacs":http://www.texmacs.org/tmweb/home/welcome.en.html is a free scientific text editor, which was both inspired by TeX and GNU Emacs. The editor allows you to write structured documents via ...
 Test
 \begin{axiom}
integrate((atan(x))^n,x)
\end{axiom}
 Test12
 FriCAS
\begin{axiom}
1 + 1
\end{axiom}
Maxima
\begin{maxima}
1 + 1;
\end{maxima}
Reduce
\begin{reduce}
1 + 1;
\end{reduce}
 Test14
 Test
 TexFormat0
 \begin{axiom}
 TexFormat (tex.spad)
 Ugly primes (op PRIME) in derivatives and unwanted quotes (" ") are now gone
 TexFormat1
 \begin{spad}
)abbrev domain TEX TexFormat
++ Author: Robert S. Sutor
++ Date Created: 1987 through 1992
++ Basic Operations: coerce, convert, display, epilogue,
++ tex, new, prologue, setEpilogue!, ...
 The Algebra of Complex Numbers Is Frobenius In Many Ways
 The Algebra of Complex Numbers Is Frobenius In Many Ways
 ToDelete
 Root for pages which probably should be deleted
 ToDo
 A short list of things to implement:
 Topics
 This page should serve to link (dispatch) to more specific topics
 TouchGraph
 TouchGraph is a graphical web page navigator.
 Tuples Products And Records
 On Fri Jul 1 14:47:12 0500 2005 Bill Page wrote:
 TutorialIntroductionToFriCAS
 A Tutorial Introduction To FriCAS
 TwistedSnakeRelation
 Nondegeneracy of the pairing (snake relation)
 Type Equivalence
 Type equivalence of domains in FriCAS and Aldor
 TypeTowerDemo
 This page demonstrates some features of FriCAS.
 TypedGcd
 GCD and types
U  Unevaluated Expressions
 Note: In FriCAS 'max' on expressions is currently undefined.
 Unicode
 The following Greek characters are supported as text::
 UnicodeIssues
 This page has more details about the unicode issue introduced on the FriCASIssues page.
 Units and Dimensions
 **On Friday, August 19, 2005 4:56 PM C Y wrote:**
 UnivariatePolynomial
 Section 9.83 UnivariatePolynomial
 UnrelatedSites
 The following web sites are not related to the Axiom
computer algebra system but have names that suggest that
they might. We collect them here for easy reference and
in case someone got here by ...
 UserInterface
 There are several options to interact with FriCAS:
V  VMWare
 Run any virtual machine
 VirtualBox
 How to provide FriCAS as a VirtualBox VDI file
==============================================
W  WebSite
 RemoteWikiURL: http://
 WesterProblemSet
 \documentclass{article}
\usepackage{axiom}
\usepackage{makeidx}
\makeindex
\begin{document}
\title{The Wester Problem Set}
\author{Timothy Daly}
\maketitle
\begin{abstract}
I'm starting to construct ...
 Where do variables belong?
 Currently, FriCAS allows
\begin{axiom}
(1/x)::UP(x, FRAC POLY INT)
\end{axiom}
 WikiWikiWeb
 Ward Cunningham's WikiWikiWeb is the original and still one of the largest
and most active wikis. It contains a wealth of information about wiki and
many other things.
 WindowsInstaller
 \documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\textbf{OpenAxiom Windows Installer Script}}
\author{Alfredo Portes}
\maketitle
 WindowsSetupNoBuild
 Complete Setup on Windows With No Builds Required
 Wish List
 Assumptions/provisos: FriCAS shuld be able to do simplifications
based on extra assumptions.
 WorkShopRISC2006
 will take place at the "Research Institute for Symbolic Computation":http://www.risc.unilinz.ac.at
in "Hagenberg":http://www.risc.unilinz.ac.at/about/map, Austria from Thursday, April 27, 10 am, to ...
 WorkShopRISC2007
 AXIOM Workshop 2007
Symmetric Functions
"RISCLinz":http://www.risc.unilinz.ac.at
 WorkShopRISC2008
 Aldor & Axiom Workshop 2008
part of
"RISC Summer 2008":http://www.risc.unilinz.ac.at/about/conferences/summer2008/
X  XDistributedPolynomial
 This type supports distributed multivariate polynomials
whose variables do not commute.
The coefficient ring may be noncommutative too.
However, coefficients and variables commute.
\begin{axiom}
)sh ...
 Xming
 Xming X Server
Y  Yacas
 From Yacas' web page:
Z  ZWiki
 Zwiki is the software which drives this wiki site.
It has been developed by "Joyful Systems":http://joyful.com and
"contributors":http://zwiki.org/ZwikiContributors around the world,
and is ...
