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Date: Wed, 29 Jun 2016 01:29:46 0400
Subject: [PATCH] books/bookvolbib Axiom Citations in the Literature
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Goal: Axiom Literate Programming
\index{Colin, Antoine}
\begin{chunk}{axiom.bib}
@article{Coli97,
author = "Colin, Antoine",
title = "Solving a system of algebraic equations with symmetries",
journal = "J. Pure Appl. Algebra",
volume = "117118",
pages = "195215",
year = "1997",
keywords = "axiomref",
abstract =
"Let $(F)$ be a system of $p$ polynomial equations
$F_i({\bf X}) \in k[{\bf X}]$, where $k$ is a commutative field and
${\bf X} := (X_1,\cdots,X_n)$ are indeterminates. Let $G$ be a subgroup
of $GL_n(k)$. A polynomial $P \in k[{\bf X}]$ (resp. rational function
$P \in k({\bf X})$ ) is an invariant of $G$ if and only if for all
$A \in G$ we have $A\cdot P = P$. We denote $k[{\bf X}]^G$ by (resp.
$k({\bf X})^G$) the algebra of polynomial (resp. rational function)
invariants of $G$. If $L$ is another subgroup of $GL_n(k)$ such that
$G \subset L$, $P$ is called a primary invariant of $G$ relative to $L$ if
and only if $Stab_L(P) = G$ (where $Stab_L(P)$ is the stabilizer of
$P$ in $L$).
The paper describes the algebra of the invariants of a finite group
and how to express these invariants in terms of a small number of
them, from both the CohenMacaulay algebra and the field theory points
of view. A method is proposed to solve $(F)$ by expressing it in terms of
primary invariants $\Pi_1,\cdots,\Pi_n$
(e.g. the elementary symmetric polynomials) and one
``primitive'' secondary invariant.
The main thrust of the paper is contained in the following theorem.
Let $(F)$ be a set of invariants of $G$. Let $L$ be a subgroup of
$GL_n(k)$ such that $G \subset L$ and $k({\bf X})^L$ is a purely
transcendental extension of $k_i$, let $\Pi_1,\cdots,\Pi_n$ be
polynomials such that $k({\bf X})^L = k(\Pi_1,\cdots,\Pi_n)$,
and let $\Theta \in k[{\bf X}]^G$ be a primitive polynomial invariant
of $G$ relative to $L$.
When possible, it is convenient to choose $\Theta$ to be one of the
polynomials in $(F)$. – An algorithm is given that allows each polynomial
$F_i$ to be expressed as $F_i({\bf X}) = H_i(\Pi_1,\cdots,\Pi_n,\Theta)$,
an algebraic fraction in $\Pi_1,\cdots,\Pi_n$ and a polynomial in
$\Theta$. Now let $L$ be the minimal polynomial of $\Theta$ over
$k[{\bf X}]^L$; we have
\[L({\bf X},T)=\prod_{\Theta^{'} \in L\cdot \Theta}(T\Theta^{'})
\in k[{\bf X}]^L[T]\]
(where $L$ is called a generic Lagrange resolvent).
As $k(\Pi_1,\cdots,\Pi_n)=k({\bf X})^L$, we can write
$L({\bf X},T)=H_0(\Pi_1,\cdots,\Pi_n,T)$ where $H_0$ is some
rational function. The question
$H_0(\Pi_1,\cdots,\Pi_n,\Theta)=0$ is always satisfied because
$\Theta$ is a root of $L$. Then, we solve the system of ($p=1$)
algebraic equations $H_i(\Pi_1,\cdots,\Pi_n,\Theta)=0$,
$0 \le i \le p$ for $\Pi_1,\cdots,\Pi_n,\Theta$ as indeterminates.
Theorem 1: Let $D \in k[\Pi_1,\cdots,\Pi_n]$ be the LCM of the
denominators of all the fractions $H_i$,$0 \le i \le p$ and let
$H_i^{'}=DH_i$. For every solution
$x:=(x_1,\cdots,x_n)$ of the system $(F)$:$F_i({\bf X})=0$,
$1 \le i \le p$, there exists a solution ($\pi_1,\cdots,\pi_n,\Theta$)
of the system
$(H^{'}):H_i^{'}(\Pi_1,\cdots,\Pi_n,\Theta)=0$, $0 \le i \le p$ such
that $x$ is a solution of the system
$(P_\pi):\Pi_i({\bf X})=\pi_i$, $1 \le i \le n$ , and of the equation
$\Theta({\bf X})=0$. Conversely, for any solution
$(\[i_1,\cdots,\pi_n,\theta)$ of the system $(H^{'})$ such that
$D(\pi_1,\cdots,\pi_n) \ne 0$, if $x$ is a solution of the system
$(P_\pi)$ relative to $(\pi_1,\cdots,\pi_n)$, then there exists
some $A \in L$ such that $\Theta(A\cdot x)=\theta$, and then for all
$B \in G$, $BA\cdot x$, is a solution of the system $(F)$.
A slighly more general version of this theorem is also given. The
paper then presents an algorithm that applies the theory and has been
implemented in AXIOM. It is followed by several examples."
}
\end{chunk}
\index{DiBlasio, Paolo}
\index{Temperini, Marco}
\begin{chunk}{axiom.bib}
@article{DiBl95,
author = "DiBlasio, Paolo and Temperini, Marco",
title = "Subtyping Inheritance and Its Application in Languages for
Symbolic Computation Systems",
journal = "J. Symbolic Computation",
volume = "19",
pages = "3963",
year = "1995",
paper = "DiBl95.pdf",
keywords = "axiomref",
abstract =
"Application of objectoriented programming techniques to design and
implementation of symbolic computation is investigated. We show the
significance of certain correctness problems, occurring in programming
environments based on specialization inheritance, due to use of method
redefinition and polymorphism. We propose a solution to these
problems, by defining a mechanism of subtyping inheritance and the
prototype of an objectoriented programming language for a symbolic
computation system. We devise the subtyping inheritance {\sl ESI
(Enhanced String Inheritance)} by lifting to programming language
constructs a given model of subtyping, which is established by a
monotonic (covariant) subtyping rule. Type safeness of language
instructions is proved.
The adoption of {\sl ESI} allows to model method and class
specialization in a natural way. The {\sl ESI} mechanism verifies the
type correctness of language statements by means of type checking
rules and preserves their correctness at runtime by a suitable method
lookup algorithm."
}
\end{chunk}
\index{DiBlasio, Paolo}
\index{Temperini, Marco}
\begin{chunk}{axiom.bib}
@InProceedings{DiBl97,
author = "DiBlasio, Paolo and Temperini, Marco",
title = "On subtyping in languages for symbolic computation systems",
booktitle = "Advances in the design of symbolic computation systems",
series = "Monographs in Symbolic Computation",
year = "1997",
publisher = "Springer",
pages = "164178",
keywords = "axiomref",
abstract =
"We want to define a strongly typed OOP language suitable as the
software development tool of a symbolic computation system, which
provides class structure to manage ADTs and supports multiple
inheritance to model specialization hierarchies. In this paper, we
provide the theoretical background for such a task."
}
\end{chunk}
\index{Fakler, Winfried}
\begin{chunk}{axiom.bib}
@article{Fakl97,
author = "Fakler, Winfried",
title = "On second order homogeneous linear differential equations with
Liouvillian solutions",
journal = "Theor. Comput. Sci.",
volume = "187",
number = "12",
pages = "2748",
year = "1997",
paper = "Fakl97.pdf",
keywords = "axiomref",
abstract =
"We determine all minimal polynomials for second order homogeneous
linear differential equations with algebraic solutions decomposed into
invariants and we show how easily one can recover the known conditions
on differential Galois groups [J. Kovacic, J. Symb. Comput. 2, 343
(1986; Zbl 0603.68035), M. F. Singer and F. Ulmer,
J. Symb. Comput. 16, 936, 3773 (1993; Zbl 0802.12004, Zbl
0802.12005), F.Ulmer and J. A. Weil, J. Symb. Comput. 22, 179200
(1996; Zbl 0871.12008)] using invariant theory. Applying these
conditions and the differential invariants of a differential equation
we deduce an alternative method to the algorithms given in (loc. cit.)
for computing Liouvillian solutions. For irreducible second order
equations our method determines solutions by formulas in all but three
cases."
}
\end{chunk}
\index{Jacquemard, Alain}
\index{KhechichineMourtada, F.Z.}
\index{Mourtada, A.}
\begin{chunk}{axiom.bib}
@article{Jacq97,
author = "Jacquemard, Alain and KhechichineMourtada, F.Z. and Mourtada, A.",
title = "Formal algorithms applied to the study of the cyclicity of a
generic algebraic polycycle with four hyperbolic crests",
journal = "Nonlinearity",
volume = "10",
number = "1",
pages = "1953",
year = "1997",
keywords = "axiomref",
comment = "french",
abstract =
"Drawing on the work of Mourtada, we show that a family of vector
fields with a generic algebraic polycycle of four hyperbolic apices
possesses a maximum capacity of four limit cycles. This cyclicity is
attained in an opening connecting the parameters which the edge
contains, in particular a generic line of singularities of dovetail
type. We also give an asymptotic estimation of the volume of this
opening, as well as an explicit example of a family of polynomial
vector fields replicating the abovedescribed conditions and
possessing five limit cycles. The methods employed are very diverse:
geometrical arguments (Thom’s theory of catastrophes and the theory of
algebraic singularities), developments from Puiseux, the number of
major roots by Descartes’ law and calculated exactly by Sturm series,
and other specific methods for formal calculus, such as for example
the cylindrical algebraic decomposition and the resolution of
algebraic systems via the construction of Gröbner bases. The
calculations have been executed formally, that is to say without
making the least appeal to numerical approximation, in using the
formal calculus system AXIOM."
}
\end{chunk}
\index{Lambe, Larry A.}
\index{Radford, David E.}
\begin{chunk}{axiom.bib}
@book{Lamb97,
author = "Lambe, Larry A. and Radford, David E.",
title = "Introduction to the quantum YangBaxter equation and quantum
groups: an algebraic approach",
booktitle = "Mathematics and its Applications",
publisher = "Kluwer Adademic Publishers",
year = "1997",
keywords = "axiomref",
abstract =
"The quantum YangBaxter equation (QYBE) has roots in statistical
mechanics and the inverse scattering method and leads to a natural
construction of a bialgebra. It turns out to have important
connections with knot theory and invariants of 3manifolds. There are
now available many reference books to quantum groups and these various
applications. The book under review develops the algebraic
underpinning and theory of the QYBE, including the constant form and
the one and two parameter forms.
We give a brief description of the chapters. Chapter 1 (together with
an Appendix) gives the algebraic preliminaries involving coalgebras,
bialgebras, Hopf algebras, modules and comodules. Chapter 2 introduces
the various forms of the QYBE, and the basic algebraic structures
associated to them, including FaddeevReshetikhinTakhtadzhan (FRT)
construction. Chapter 3 explores various categorical settings for the
constant form of the QYBE, the most basic being the category of left
QYB modules over a bialgebra and the notion of algebras, coalgebras,
etc. in this category. Chapter 4 develops universal mapping properties
of the FRT construction and its reduced version, and the authors
investigate when the reduced FRT construction leads to a pointed
bialgebra or a pointed Hopf algebra. Chapter 5 develops the quantum
groups associated to $SL(2)$, i.e., the quantum universal enveloping
algebra, and the quantum function algebra. Chapter 6 introduces
quasitriangular Hopf algebras, and discusses how the
finitedimensional ones give rise to solutions of the QYBE through
their representation theory. The most important example is the
Drinfeld double of a finitedimensional Hopf algebra. The authors note
(through an exercise!) that every finitedimensional Hopf algebra is
the reduced FRT construction of some solution to the QYBE. Chapter 7
introduces coquasitriangular bialgebras, the most important being the
FRT and the reduced FRT constructions. There are some generalizations
here to the oneparameter form of the QYBE. Chapter 8 uses all the
previously developed techniques to find solutions of the QYBE in
certain cases, including the oneparameter form. Some of these were
discovered by computer algebra methods. The final chapter 9 gives a
brief discussion of certain categorical constructions and the QYBE is
certain fairly abstract categories, motivated by the fact that the FRT
construction is a coend.
This book fills an important niche in the literature involving the
QYBE by highlighting the algebraic aspects and applications. Although
this is basically a reference book, it includes so many important
parts of the study of Hopf algebras that it could be used as a
textbook for a certain type of course on Hopf algebras and quantum
groups, and certainly as supplementary reading material for such a
course. There are frequent exercises which would be useful for such
purposes. Besides being a basic source book, the authors include some
new results and some novel approaches to earlier results. All this
makes this book a most welcome addition to the quantum group
literature."
}
\end{chunk}
\index{Letichevskij, A. Alexander}
\index{Marinchenko, V. G.}
\begin{chunk}{axiom.bib}
@article{Leti97,
author = "Letichevskij, A. Alexander and Marinchenko, V. G.",
title = "Objects in algebraic programming system",
journal = "Cybern. Syst. Anal.",
volume = "33",
number = "2",
pages = "283299",
year = "1997",
keywords = "axiomref",
comment = "translated from Russian",
abstract =
"The algebraic programming system (APS) developed at the
V. M. Glushkov Institute of Cybernetics of the Academy of Sciences of
the Ukrainian SSR integrates the basic programming paradigms,
including procedural, functional, algebraic, and logic programming.
Algebraic programming in APS relies on special data structures, the
socalled graph terms, which permit using diverse data and knowledge
representations in relevant application domains. In the language
APLAN, graph terms are described by expressions or systems of
expressions of a manysorted algebra of data. They may represent both
objects of the application domain and reasoning about these
objects. The option of setting an arbitrary interpretation of the
operations in the algebra of data makes it possible to use APS as a
basis for various extensions.
Symbolic computation systems such as Scratchpad/AXIOM have acquired
special importance. They provide various possibilities of manipulating
typed mathematical objects, including objects of complex hierarchical
structure. This is a natural requirement when working with algebraic
objects. In particular, the properties of many algebraic structures
(such as groups, rings, fields, etc.) are naturally
hierarchicalmodular.
The Institute of Cybernetics and the Kherson Teachers’ College have
developed an instructionoriented computer algebra system AIST. The
AIST kernel is a hierarchical structure of mathematical concepts
described in the APS language. However, construction of new
applications on the basis of this hierarchical structure has proved
difficult. The system kernel can be made more flexible by providing
tools for flexible description of hierarchical structures of
mathematical concepts.
In this article, we describe an extension of the language APLAN, which
provides tools for the objectoriented style of programming. This is
one of the possible ways of introducing types in APS. The
objectoriented technology also can be used to develop a hierarchical
system of mathematical objects."
}
\end{chunk}
\index{Schwarzweller, Christoph}
\begin{chunk}{axiom.bib}
@phdthesis{Schw97,
author = "Schwarzweller, Christoph",
title = "MIZAR verification of generic algebraic algorithms",
school = "University of Tubingen",
year = "1997",
paper = "Schw97.pdf",
keywords = "axiomref",
abstract =
"Although generic programming founds more and more attention –
nowadays generic programming languages as well as generic libraries
exist – there are hardly approaches for the verification of generic
algorithms or generic libraries. This thesis deals with generic
algorithms in the field of computer algebra. We propose the Mizar
system as a theorem prover capable of verifying generic algorithms on
an appropriate abstract level. The main advantage of the MIZAR theorem
prover is its special input language that enables textbook style
presentation of proofs. For generic versions of Brown/Henrici addition
and of Euclidean’s algorithm we give complete correctness proofs
written in the MIZAR language.
Moreover, we do not only prove algorithms correct in the usual
sense. In addition we show how to check, using the MIZAR system, that
a generic algebraic algorithm is correctly instantiated with a
particular domain. Answering this question that especially arises if
one wants to implement generic programming languages, in the field of
computer algebra requires nontrivial mathematical knowledge.
To build a verification system using the MIZAR theorem prover, we also
implemented a generator which almost automatically computes for a
given algorithm a set of theorems that imply the correctness of this
algorithm."
}
\end{chunk}
\index{Zenger, Christoph}
\begin{chunk}{axiom.bib}
@article{Zeng97,
article = "Zenger, Christoph",
title = "Indexed types",
journal = "Theor. Comput. Sci.",
volume = "187",
numbers = "12",
pages = "147165",
year = "1997",
keywords = "axiomref",
paper = "Zeng97.pdf",
abstract =
"A new extension of the Hindley/Milner type system is proposed. The
type system has algebraic types, that have not only type parameters
but also value parameters (indices). This allows for example to
parameterize matrices and vectors by their size and to check size
compatibility statically. This is especially of interest in computer
algebra."
}
\end{chunk}
\index{Bernardin, Laurent}
\begin{chunk}{axiom.bib}
@article{Bern96,
author = "Benardin, Laurent",
title = "A review of symbolic solvers",
journal = "SIGSAM Bull.",
volume = "30",
number = "1",
pages = "920",
year = "1996",
keywords = "axiomref",
paper = "Bern96.pdf",
abstract =
"Solving equations and systems of equations symbolically is a key
feature of every Computer Algebra System. This review examines the
capabilities of the six best known general purpose systems to date in
the area of general algebraic and transcendental equation
solving. Areas explicitly not covered by this review are differential
equations and numeric or polynomial system solving as special purpose
systems exist for these kinds of problems. The aim is to provide a
benchmark for comparing Computer Algebra Systems in a specific
domain. We do not intend to give a rating of overall capabilities as
for example in [9]. 1 The Contestants We compare six major Computer
Algebra Systems. Axiom 2.0 [7], Derive 3.06 [1], Macsyma 420 [8],
Maple V R4 [3], Mathematica 2.2 [10], MuPAD 1.2.9 [5] and Reduce 3.6
[6]. When available, we tried to use the latest shipping version of
each system. 2 The Problem Set The following table presents the set of
80 problems that we used to evaluate the different solvers..."
}
\end{chunk}
\index{Wester, Michael J.}
\begin{chunk}{axiom.bib}
@misc{Westxx,
author = "Wester, Michael J.",
title = "Computer Algebra Synonyms",
keywords = "axiomref",
url = "http://math.unm.edu/~wester/cas/synonyms.pdf",
paper = "Westxx.pdf",
abstract =
"The following is a collection of synonyms for various operations in
the seven general purpose computer algebra systems {\bf Axiom}, {\bf
Derive}, {\bf Macsyma}, {\bf Maple}, {\bf Mathematica}, {\bf MuPAD},
and {\bf Reduce}. This collection does not attempt to be
comprehensive, but hopefully it will be useful in giving an indication
of how to translate between the syntaxes used by the different systems
in many common situations. Note that for a blank entry means that
there is no exact translation of a particular operation for the
indicated system, but it may still be possible to work around this
lack with a related functionality."
}
\end{chunk}
\index{Wester, Michael J.}
\begin{chunk}{axiom.bib}
@misc{West95,
author = "Wester, Michael J.",
title = "A Review of CAS Mathematical Capabilities",
year = "1995",
keywords = "axiomref",
paper = "West95.pdf",
url = "http://math.unm.edu/~wester/cas/Paper.ps",
abstract =
"Computer algebra systems (CASs) have become an important
computational tool in the last decade. General purpose CASs, which are
designed to solve a wide variety of problems, have gained special
prominance. In this paper, the capabilities of seven major general
purpose CASs (Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, and
Reduce) are reviewed on 131 short problems covering a broad range of
(primarily) symbolic mathematics.
A demo was developed for each CAS, run and the results
evaluated. Problems were graded in terms of whether it was easy or
difficult or possible to produce an answer and if an answer was
produced, whether it was correct. It is the author's hope that this
review will encourage the development of a comprehensive CAS test
suite."
}
\end{chunk}
\index{Apel, Joachim}
\index{Klaus, Uwe}
\begin{chunk}{axiom.bib}
@misc{Apel94,
author = "Apel, Joachim and Klaus, Uwe",
title = "Representing Polynomials in Computer Algebra Systems",
year = "1994",
paper = "Apel94.pdf",
abstract =
"There are discussed implementational aspects of the specialpurpose
computer algebra system FELIX designed for computations in
constructive algebra. In particular, data types developed for the
representation of and computation with commutative and noncommuative
polynomials are described. Furthermore, comparison of time and memory
requirements of different polynomial representations are reported."
}
\end{chunk}
\index{Stoutemyer, David R.}
\begin{chunk}{axiom.bib}
@article{Stou91,
author = "Stoutemyer, David R.",
title = "Crimes and misdemeanors in the computer algebra trade",
journal = "Notices of the American Mathematical Society",
volume = "38",
number = "7",
pages = "778785",
year = "1991"
}
\end{chunk}
\index{Sangwin, Chris}
\begin{chunk}{axiom.bib}
@misc{Sang10,
author = "Sangwin, Chris",
title = "Intriguing Integrals: Part I and II",
year = "2010",
url1 =
"https://plus.maths.org/issue54/features/sangwin/2pdf/index.html/op.pdf",
paper1 = "Sang10a.pdf",
url2 =
"https://plus.maths.org/issue54/features/sangwin2/2pdf/index.html/op.pdf",
paper2 = "Sang10b.pdf"
}
\end{chunk}
\index{Evans, Brian}
\begin{chunk}{axiom.bib}
@misc{Evanxx,
author = "Evans, Brian",
title = "History of CA Systems",
url = "http://felix.unife.it/Root/dMathematics/dThemathematician/dHistoryofmathematics/tHistoryofcomputeralgebra",
paper = "Evanxx.txt"
}
\end{chunk}
\index{Martin, Ursula}
\index{Shand, D.}
\begin{chunk}{axiom.bib}
@misc{Mart97,
author = "Martin, Ursula and Shand, D",
title = "Investigating some Embedded Verification Techniques for
Computer Algebra Systems",
url = "http://www.risc.jku.at/conferences/Theorema/papers/shand.ps.gz",
paper = "Mart97.ps",
abstract = "
This paper reports some preliminary ideas on a collaborative project
between St. Andrews University in the UK and NAG Ltd. The project aims
to use embedded verification techniques to improve the reliability and
mathematical soundness of computer algebra systems. We give some
history of attempts to integrate computer algebra systems and
automated theorem provers and discuss possible advantages and
disadvantages of these approaches. We also discuss some possible case
studies."
}
\end{chunk}
\index{Tonisson, Eno}
\begin{chunk}{axiom.bib}
@article{Tonixx,
author = "Tonisson, Eno",
title = "Branch Completeness in School Mathematics and in Computer Algebra
Systems",
journal = "The Electronic Journal of Mathematics and Technology",
volume = "1",
number = "1",
issn = "19332823",
paper = "Tonixx.pdf",
url = "https://php.radford.edu/~ejmt/deliveryBoy.php?paper=eJMT_v1n3p5",
abstract =
"In many cases when solving school algebra problems (e.g. simplifying
an expression, solving an equation), the solution is separable into
branches in some manner. The paper describes some approaches to
branches that are used in school textbooks and computer algebra
systems and compares them with mathematically branchcomplete
solutions. It tries to identify possible reasons behind different
approaches and also indicate some ideas how such differences could be
explained to the students."
}
\end{chunk}
\index{Beeson, Michael}
\begin{chunk}{axiom.bib}
@misc{Beesxx,
author = "Beeson, Michael",
title = "Automatic Generation of EpsilonDelta Proofs of Continuity",
url = "http://www.michaelbeeson.com/research/papers/aisc.pdf",
paper = "Beesxx.pdf",
abstract =
"As part of a project on automatic generation of proofs involving both
logic and computation, we have automated the production of some proofs
involving epsilondelta arguments. These proofs involve two or three
quantifiers on the logical side, and on the computational side, they
involve algebra, trigonometry, and some calculus. At the border of
logic and computation, they involve several types of arguments
involving inequalities, including transitivity chaining and several
types of bounding arguments, in which bounds are sought that do not
depend on certain variables. Control mechanisms have been developed
for intermixing logical deduction steps with computational steps and
with inequality reasoning. Problems discussed here as examples involve
the continuity and uniform continuity of various specific functions."
}
\end{chunk}
\index{Ballarin, Clemens}
\index{Paulson, Lawrence C.}
\begin{chunk}
@misc{Ball98,
author = "Ballarin, Clemens and Paulson, Lawrence C.",
title = "Reasoning about Coding Theory: The Benefits We Get from
Computer Algebra",
year = "1998",
url = http://www21.in.tum.de/~ballarin/publications/aisc98.pdf",
paper = "Ball98.pdf",
abstract =
"The use of computer algebra is usually considered beneficial for
mechanised reasoning in mathematical domains. We present a case study,
in the application domain of coding theory, that supports this claim:
the mechanised proof depends on nontrivial algorithms from computer
algebra and increase the reasoning power of the theorem prover. The
unsoundness of computer algebra systems is a major problem in
interfacing them to theorem provers. Our approach to obtaining a sound
overall system is not blanket distrust but based on the distinction
between algorithms we call sound and {\sl ad hoc} respectively. This
distinction is blurred in most computer algebra systems OUr
experimental interface therefore uses a computer algebra library. It
is based on theorem templates, which provide formal specifications for
the algorithms."
}
\end{chunk}
\index{Aslaksen, Helmer}
\begin{chunk}{axiom.bib}
@article{Asla96,
author = "Aslaksen, Helmer",
title = "Multiplevalued complex functions and computer algebra",
journal = "SIGSAM Bulletin",
volume = "30",
number = "2",
year = "1996",
pages = "1220",
paper = "Asla96.pdf",
url = "http://www.math.nus.edu.sg/aslaksen/papers/cacas.pdf",
abstract =
"I recently taught a course on complex analysis. That forced me to
think more carefully about branches. Being interested in computer
algebra, it was only natural that I wanted to see how such programs
dealt with these problems. I was also inspired by a paper by
Stoutemyer.
While programs like Derive, Maple, Mathematica and Reduce are very
powerful, they also have their fair share of problems. In particular,
branches are somewhat of an Achilles' heel for them. As is wellknown,
the complex logarithm function is properly defined as a
multiplevalued function. And since the general power and exponential
functions are defined in terms of the logarithm function, they are
also multiplevalued. But for actual computations, we need to make
them single valued, which we do by choosing a branch. In Section 2, we
will consider some transformation rules for branches of
multiplevalued complex functions in painstaking detail.
The purpose of this short article is not to do a comprehensive
comparative study of different computer algebra systems. My goal is
simply to make the readers aware of some of the problems, and to
encourage the readers to sit down and experiment with their favourite
programs."
}
\end{chunk}
\index{Fateman, Richard J.}
\begin{chunk}{axiom.bib}
@InProceedings{Fate96,
author = "Fateman, Richard J.",
title = "A Review of Symbolic Solvers",
booktitle = "Proc 1996 ISSAC",
series = "ISSAC 96",
year = "1996",
pages = "8694",
keywords = "axiomref",
keywords = "axiomref",
paper = "Fate96.pdf",
url = "http://http.cs.berkeley.edu/~fateman/papers/eval.ps",
abstract =
"``Evaluation'' of expressions and programs in a computer algebra
system is central to every system, but inevitably fails to provide
complete satisfaction. Here we explain the conflicting requirements,
describe some solutions from current systems, and propose alternatives
that might be preferable sometimes. We give examples primarily from
Axiom, Macsyma, Maple, Mathematica, with passing metion of a few other
systems."
}
\end{chunk}
\index{Fateman, Richard J.}
\begin{chunk}{axiom.bib}
@misc{Fate05,
author = "Fateman, Richard J.",
title = "An incremental approach to building a mathematical
expert out of software",
conference = "Axiom Computer Algebra Conference",
location = "City College of New York, CAISS project",
year = "2005",
month = "April",
day = "19",
url = "http://www.cs.berkeley.edu/~fateman/papers/axiom.pdf",
paper = "Fat05.pdf",
keywords = "axiomref"
}
\end{chunk}
\index{Gr\"abe, HansGert}
\begin{chunk}{axiom.bib}
@misc{Grab98,
author = "Grabe, HansGert",
title = "About the Polynomial System Solve Facility of Axiom, Macsyma,
Maple Mathematica, MuPAD, and Reduce",
paper = "Grab98.pdf",
url =
"https://www.informatik.unileipzig.de/~graebe/ComputerAlgebra/Publications/WesterBook.pdf",
keywords = "axiomref",
abstract =
"We report on some experiences with the general purpose Computer
Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and
Reduce solving systems of polynomial equations and the way they
present their solutions. This snapshot (taken in the spring of 1996)
of the current power of the different systems in a special area
concentrates on both CPUtimes and the quality of the output."
}
\end{chunk}
\index{Gr\"abe, HansGert}
\begin{chunk}{axiom.bib}
@misc{Grab06,
author = "Grabe, HansGert",
title = "The Groebner Factorizer and Polynomial System Solving",
year = "2006",
keywords = "axiomref",
report = "Special Semester on Groebner Bases",
location = "Linz",
paper = "Grab06.pdf",
url =
"https://www.ricam.oeaw.ac.at/specsem/srs/groeb/download/06\_02\_Solver.pdf",
abstract =
"Let $S := k[x_1,\ldots, x_n]$ be the polynomial ring in the
variables $x_1,\ldots,x_n$ over the field $k$ and
$B := \{f_1,\ldots,f_m\} \subset S$
be a finite system of polynomials. Denote by $I(B)$ the
ideal generated by these polynomials. One of the major tasks of
constructive commutative algebra is the derivation of information
about the structure of
\[V(B):=\{a \in K^n : \forall f \in B{\rm\ such\ that\ }f(a)=0\}\]
the set of common zeroes of the system $B$ over an
algebraically closed extension $K$ of $k$. Splitting the system into
smaller ones, solving them separately, and patching all solutions
together is often a good guess for a quick solution of even highly
nontrivial problems. This can be done by several techniques, e.g.,
characteristic sets, resultants, the Groebner factorizer or some ad
hoc methods. Of course, such a strategy makes sense only for problems
that really will split, i.e., for reducible varieties of
solutions. Surprisingly often, problems coming from 11real life''
fulfill this condition.
Among the methods to split polynomial systems into smaller pieces
probably the Groebner factor izer method attracted the most
theoretical attention, see Czapor ([4, 5]), Davenport ([6]), Melenk, M
̈oller and Neun ([16, 17]) and Gr ̈abe ([13, 14]). General purpose
Computer Algebra Systems (CAS) are well suited for such an approach,
since they make available both a (more or less) well tuned
implementation of the classical Groebner algorithm and an effective
multivariate polynomial factorizer.
Furthermore it turned out that the Groebner factorizer is not only a
good heuristic approach for splitting, but its output is also usually
a collection of almost prime components. Their description allows a
much deeper understanding of the structure of the set of zeroes
compared to the result of a sole Groebner basis computation.
Of course, for special purposes a general CAS as a multipurpose
mathematical assistant can’t offer the same power as specialized
software with efficiently implemented and well adapted algorithms and
data types. For polynomial system solving, such specialized software
has to implement two algorithmically complex tasks, solving and
splitting, and until recently none of the specialized systems (as
e.g., GB, Macaulay, Singular, CoCoA, etc.) did both
efficiently. Meanwhile, being very efficient computing (classical)
Groebner bases, development efforts are also directed, not only
for performance reasons, towards a better inclusion of factorization
into such specialized systems. Needless to remark that it needs some
skill to force a special system to answer questions and the user will
probably first try his ``home system'' for an answer. Thus the
polynomial systems solving facility of the different CAS should behave
especially well on such polynomial systems that are hard enough not to
be done by hand, but not really hard to require special efforts. It
should invoke a convenient interface to get the solutions in a form
that is (correct and) well suited for further analysis in the familiar
environment of the given CAS as the personal mathematical assistant."
}
\end{chunk}
\index{Corless, Robert M.}
\index{Jeffrey, David J.}
\index{Watt, Stephen M.}
\index{Bradford, Russell}
\index{Davenport, James H.}
\begin{chunk}{axiom.bib}
@misc{Corl0,
author = "Corless, Robert M. and Jeffrey, David J. and Watt, Stephen M.
and Bradford, Russell and Davenport, James H.",
title = "Reasoning about the elementary functions of complex analysis",
url = "http://www.csd.uwo.ca/~watt/pub/reprints/2002amaireasoning.pdf",
paper = "Corl05.pdf",
abstract = "
There are many problems with the simplification of elementary
functions, particularly over the complex plane. Systems tend to make
``howlers'' or not to simplify enough. In this paper we outline the
``unwinding number'' approach to such problems, and show how it can be
used to prevent errors and to systematise such simplification, even
though we have not yet reduced the simplification process to a
complete algorithm. The unsolved problems are probably more amenable
to the techniques of artificial intelligence and theorem proving than
the original problem of complexvariable analysis."
}
\end{chunk}
\index{Touratier, Emmanuel}
\begin{chunk}{axiom.bib}
@misc{Tour98,
author = "Touratier, Emmanuel",
title = {Etude du typage dans le syst\`eme de calcul scientifique Aldor},
comment = "Study of types in the Aldor scientific computation system",
year = "1998",
paper = "Tour98.pdf",
url = "http://axiomwiki.newsynthesis.org/public/refs/AldorT1998_04.pdf",
keywords = "axiomref"
}
\end{chunk}
\index{Seiler, Werner Markus}
\begin{chunk}{axiom.bib}
@misc{Seil95,
author = "Seiler, Werner Markus",
title = "Applying AXIOM to partial differential equations",
institution = {Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik},
year = "1995",
type = "Internal Report",
number = "9517",
url = "http://axiomwiki.newsynthesis.org/public/refs/Axiompdf.pdf",
paper = "Seil95.pdf",
keywords = "axiomref",
abstract =
"We present an Axiom environment called JET for geometric computations
with partial differential equations within the framework of the jet
bundle formalism. This comprises expecially the completion of a given
differential equation to an involutive one according to the
CartanKuranishi Theorem and the setting up of the determining system
for the generators of classical and nonclassical Lie
symmetries. Details of the implementations are described and
applications are given. An appendix contains tables of all exported
functions."
}
\end{chunk}
\index{Davenport, James H.}
\begin{chunk}{axiom.bib}
@misc{Dave84a,
author = "Davenport, James H.",
title = "A New Algebra System",
paper = "Dave84a.pdf",
keywords = "axiomref",
url = "http://axiomwiki.newsynthesis.org/public/refs/Davenport1984a\_new\_algebra\_system.pdf",
abstract =
"Seminal internal paper discussing Axiom design decisions."
}
\end{chunk}
\index{Conrad, Marc}
\index{French, Tim}
\index{Maple, Carsten}
\index{Pott, Sandra}
\begin{chunk}{axiom.bib}
@misc{Conrxxa,
author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra",
title = "Approaching Inheritance from a Natural Mathematical Perspective
and from a Java Driven Viewpoint: a Comparative Review",
keywords = "axiomref",
url = "http://axiomwiki.newsynthesis.org/public/refs/McTfCmSpaxiom.pdf",
paper = "Conrxxa.pdf",
abstract = "
It is wellknown that few objectoriented programming languages allow
objects to change their nature at runtime. There have been a number
of reasons presented for this, but it appears that there is a real
need for matters to change. In this paper we discuss the need for
objectoriented programming languages to reflect the dynamic nature of
problems, particularly those arising in a mathematical context. It is
from this context that we present a framework that realistically
represents the dynamic and evolving characteristic of problems and
algorithms."
}
\end{chunk}
\index{Meijer, Erik}
\index{Fokkinga, Maarten}
\index{Paterson, Ross}
\begin{chunk}{axiom.bib}
@misc{Meij91,
author = "Meijer, Erik and Fokkinga, Maarten and Paterson, Ross",
title = "Functional Programming with Bananas, Lenses, Envelopes and
Barbed Wire",
url = "http://eprints.eemcs.utwente.nl/7281/01/dbutwente40501F46.pdf",
paper = "Meij91.pdf",
abstract = "
We develop a calculus for lazy functional programming based on
recursion operators associated with data type definitions. For these
operators we derive various algebraic laws that are useful in deriving
and manipulating programs. We shall show that all example functions in
Bird and Wadler's ``Introduction to Functional Programming'' can be
expressed using these operators."
}
\end{chunk}
\index{Robidoux, Nicolas}
\begin{chunk}{axiom.bib}
@misc{Robi93,
author = "Robidoux, Nicolas",
title = "Does Axiom Solve Systems of O.D.E's Like Mathematica?",
year = "1993",
paper = "Robi93.pdf",
url = "http://axiomwiki.newsynthesis.org/public/refs/Robidoux.pdf",
keywords = "axiomref",
abstract = "
If I were demonstrating Axiom and were asked this question, my reply
would be ``No, but I am not sure that this is a bad thing''. And I
would illustrate this with the following example.
Consider the following system of O.D.E.'s
\[
\begin{array}{rcl}
\frac{dx_1}{dt} & = & \left(1+\frac{cos t}{2+sin t}\right)x_1\\
\frac{dx_2}{dt} & = & x_1  x_2
\end{array}
\]
This is a very simple system: $x_1$ is actually uncoupled from $x_2$"
}
\end{chunk}
\index{Davenport, James H.}
\index{Faure, Christ\'ele}
\begin{chunk}{axiom.bib}
@misc{Davexx,
author = {Davenport, James; Faure, Christ\'ele},
title = "The Unknown in Computer Algebra",
url =
"http://axiomwiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf",
paper = "Davexx.pdf",
keywords = "axiomref",
abstract = "
Computer algebra systems have to deal with the confusion between
``programming variables'' and ``mathematical symbols''. We claim that
they should also deal with ``unknowns'', i.e. elements whose values
are unknown, but whose type is known. For examples $x^p \ne x$ if $x$
is a symbol, but $x^p = x$ if $x \in GF(p)$. We show how we have
extended Axiom to deal with this concept."
}
\end{chunk}
\index{Davenport, James H.}
\begin{chunk}{axiom.bib}
@techreport{Dave92b,
author = "Davenport, James H.",
title = "How does one program in the AXIOM system?",
institution = "Numerical Algorithms Group, Inc.",
year = "1992",
type = "technical report",
number = "TR6/92 (ATR/4)(NP2493)",
url = "http://www.nag.co.uk/doc/TechRep/axiomtr.html",
paper = "Dave92b.pdf",
keywords = "axiomref",
abstract =
"Axiom is a computer algebra system superficially like many others, but
fundamentally different in its internal construction, and therefore in
the possibilities it offers to its users and programmers. In these
lecture notes, we will explain, by example, the methodology that the
author uses for programming substantial bits of mathematics in Axiom."
}
\end{chunk}
\index{Youssef, Saul}
\begin{chunk}{axiom.bib}
@misc{Yous04,
author = "Youssef, Saul",
title = "Prospects for Category Theory in Aldor",
year = "2004",
url =
"http://axiomwiki.newsynthesis.org/public/refs/YoussefProspectsForCategoryTheoryInAldor.pdf",
paper = "Yous04.pdf",
abstract =
"Ways of encorporating category theory constructions and results into
the Aldor language are discussed. The main features of Aldor which
make this possible are identified, examples of categorical
constructions are provided and a suggestion is made for a foundation
for rigorous results."
}
\end{chunk}
\index{Carpent, Quentin}
\index{Conil, Christophe}
\begin{chunk}{axiom.bib}
@misc{Carp04,
author = "Carpent, Quentin and Conil, Christophe",
title = "Utilisation de logiciels libres pour la r\'ealisation de TP MT26",
year = "2004",
paper = "Carp04.pdf",
url = "http://axiomwiki.newsynthesis.org/public/refs/ac20.pdf",
keywords = "axiomref",
comment = "french",
abstract = "radicalSolve(x**3+x**27=0,x)"
}
\end{chunk}
\index{Naylor, William A.}
\index{Padget, Julian}
\begin{chunk}{axiom.bib}
@InProceedings{Nayl06,
author = "Naylor, William and Padget, Julian",
title = "From Untyped to Polymorphically Typed Objects in Mathematical
Web Services",
paper = "NPxx.pdf",
series = Lecture Notes in Computer Science",
volume = "4108",
pages = "222236",
year = "2006",
keywords = "axiomref",
abstract =
"OpenMath is a widely recognized approach to the semantic markup of
mathematics that is often used for communication between OpenMath
compliant systems. The Aldor language has a sophisticated
categorybased type system that was specifically developed for the
purpose of modelling mathematical structures, while the system itself
supports the creation of smallfootprint applications suitable for
deployment as web services. In this paper we present our first results
of how one may perform translations from generic OpenMath objects into
values in specific Aldor domains, describing how the Aldor interfae
domain ExpresstionTree is used to achieve this. We outline our Aldor
implementation of an OpenMath translator, and describe an efficient
extention of this to the Parser category. In addition, the Aldor
service creation and invocation mechanism are explained. Thus we are
in a position to develop and deploy mathematical web services whose
descriptions may be directly derived from Aldor's rich type language."
}
\end{chunk}
\index{Watt, Stephen M.}
\index{Broadbery, Peter A.}
\index{Dooley, Sam}
\index{Iglio, Pietro}
\begin{chunk}{axiom.bib}
@techreport{Watt94,
author = "Watt, Stephen M. and Broadbery, Peter A. and Dooley, Samuel S.
and Iglio, Pietro",
title = "A First Report on the A\# Compiler (including benchmarks)",
institution = "IBM Research",
year = "1994",
type = "technical report",
number = "RC19529 (85075)",
paper = "Watt94.pdf",
url =
"http://axiomwiki.newsynthesis.org/public/refs/axiomaldorasharp.pdf",
keywords = "axiomref",
abstract =
"The $A^{#}$ compiler allows users of computer algebra to develop
programs in a context where multiple programming languages are
employed. The compiler translates programs written in the $A^{#}$
programming language to a lowlevel intermediate language, Foam,
from which it can generate standalone programs, native object
libraries to be linked with other applications, or code to be read
into closed environments. In addition, Foam code may be directly
executed using an interpreter provided with the $A^{#}$ compiler.
The $A^{#}$ programming language provides support for objectoriented
and functional programming styles. It is ``higherorder'' in the sense
that both types and functions are first class, and may be manipulated
in the same ways as any other values. The primary considerations in
the formulation of the language have been generality, composibility,
and efficiency. The language has been designed to admit a number of
important optimizations, allowing compilation to machine code which is
in many instances of efficiency comparable to that produced by a C or
Fortran compiler.
The original motivation for $A^{#}$ comes from the field of computer
algebra: to provide an improved extension language for the Axiom
computer algebra system."
}
\end{chunk}
\index{Lambe, Larry A.}
\index{Luczak, Richard}
\begin{chunk}{axiom.bib}
@article{Lamb93a,
article = "Lambe, Larry and Luczak, Richard",
title = "ObjectOriented Mathematical Programming and
Symbolic/Numeric Interface",
journal = "3rd Int. Conf. on Expert Systems in Numerical Computing",
year = "1993",
url = "http://axiomwiki.newsynthesis.org/public/refs/axiomfem.pdf",
paper = "Lamb93a.pdf",
keywords = "axiomref",
abstract =
"The Axiom language is based on the notions of ``categories'',
``domains'', and ``packages''. These concepts are used to build an
interface between symbolic and numeric calculations. In particular, an
interface to the NAG Fortran Library and Axiom's algebra and graphics
facilities is presented. Some examples of numerical calculations in a
symbolic computational environment are also included using the finite
element method. While the examples are elementary, we believe that
they point to very powerful methods for combining numeric and symbolic
computational techniques."
}
\end{chunk}
\index{Griesmer, James H.}
\index{Jenks, Richard D.}
\begin{chunk}{axiom.bib}
@InProceedings{Grie71,
author = "Griesmer, James H. and Jenks, Richard D.",
title = "SCRATCHPAD/1  an interactive facility for symbolic mathematics",
booktitle = "Proc. second ACM Symposium on Symbolic and Algebraic
Manipulation",
series = "SYMSAC 71",
year = "1971",
pages = "4258",
url = "http://delivery.acm.org/10.1145/810000/806266/p42griesmer.pdf",
paper = "GJ71.pdf",
keywords = "axiomref",
abstract = "
The SCRATCHPAD/1 system is designed to provide an interactive symbolic
computational facility for the mathematician user. The system features
a user language designed to capture the style and succinctness of
mathematical notation, together with a facility for conveniently
introducing new notations into the language. A comprehensive system
library incorporates symbolic capabilities provided by such systems as
SIN, MATHLAB, and REDUCE."
}
\end{chunk}
\index{Seiler, Werner Markus}
\index{Calmet, J.}
\begin{chunk}{axiom.bib}
@misc{Seil95a,
author = "Seiler, Werner Markus and Calmet, J.",
title = "JET  An Axiom Environment for Geometric Computations with
Differential Equations",
paper = "Seil95a.pdf",
url = "http://axiomwiki.newsynthesis.org/public/refs/axiomjet95.pdf",
keywords = "axiomref",
abstract =
"JET is an environment within the computer algebra system Axiom to
perform such computations. The current implementation emphasises the
two key concepts involution and symmetry. It provides some packages
for the completion of a given system of differential equations to an
equivalent involutive one based on the CartanKuranishi theorem and
for setting up the determining equations for classical and
nonclassical point symmetries."
}
\end{chunk}

books/bookvolbib.pamphlet  1117 +++++++++++++++++++++++++++++++++
changelog  2 +
patch  1189 +++++++++++++++++++++++++++++++++++++++
src/axiomwebsite/patches.html  2 +
4 files changed, 2186 insertions(+), 124 deletions()
diff git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet
index 88a05d7..8e72be5 100644
 a/books/bookvolbib.pamphlet
+++ b/books/bookvolbib.pamphlet
@@ 3389,8 +3389,8 @@ when shown in factored form.
\index{Watt, Stephen M.}
\index{Bradford, Russell}
\index{Davenport, James H.}
\begin{chunk}{ignore}
{Corl0,
+\begin{chunk}{axiom.bib}
+@misc{Corl0,
author = "Corless, Robert M. and Jeffrey, David J. and Watt, Stephen M.
and Bradford, Russell and Davenport, James H.",
title = "Reasoning about the elementary functions of complex analysis",
@@ 3725,9 +3725,11 @@ Mathematics of Computation, Vol 32, No 144 Oct 1978, pp12151231
\index{Meijer, Erik}
\index{Fokkinga, Maarten}
\index{Paterson, Ross}
\begin{chunk}{ignore}
\bibitem[Meijer 91]{Meij91} Meijer, Erik; Fokkinga, Maarten; Paterson, Ross
 title = "Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire",
+\begin{chunk}{axiom.bib}
+@misc{Meij91,
+ author = "Meijer, Erik and Fokkinga, Maarten and Paterson, Ross",
+ title = "Functional Programming with Bananas, Lenses, Envelopes and
+ Barbed Wire",
url = "http://eprints.eemcs.utwente.nl/7281/01/dbutwente40501F46.pdf",
paper = "Meij91.pdf",
abstract = "
@@ 3737,21 +3739,26 @@ Mathematics of Computation, Vol 32, No 144 Oct 1978, pp12151231
and manipulating programs. We shall show that all example functions in
Bird and Wadler's ``Introduction to Functional Programming'' can be
expressed using these operators."
+}
\end{chunk}
\index{Youssef, Saul}
\begin{chunk}{ignore}
\bibitem[Youssef 04]{You04} Youssef, Saul
+\begin{chunk}{axiom.bib}
+@misc{Yous04,
+ author = "Youssef, Saul",
title = "Prospects for Category Theory in Aldor",
year = "2004",
 paper = "You04.pdf",
 abstract = "
 Ways of encorporating category theory constructions and results into
+ url =
+"http://axiomwiki.newsynthesis.org/public/refs/YoussefProspectsForCategoryTheoryInAldor.pdf",
+ paper = "Yous04.pdf",
+ abstract =
+ "Ways of encorporating category theory constructions and results into
the Aldor language are discussed. The main features of Aldor which
make this possible are identified, examples of categorical
constructions are provided and a suggestion is made for a foundation
for rigorous results."
+}
\end{chunk}
@@ 3819,6 +3826,58 @@ Martin, U.
\end{chunk}
+\index{Ballarin, Clemens}
+\index{Paulson, Lawrence C.}
+\begin{chunk}
+@misc{Ball98,
+ author = "Ballarin, Clemens and Paulson, Lawrence C.",
+ title = "Reasoning about Coding Theory: The Benefits We Get from
+ Computer Algebra",
+ year = "1998",
+ url = http://www21.in.tum.de/~ballarin/publications/aisc98.pdf",
+ paper = "Ball98.pdf",
+ abstract =
+ "The use of computer algebra is usually considered beneficial for
+ mechanised reasoning in mathematical domains. We present a case study,
+ in the application domain of coding theory, that supports this claim:
+ the mechanised proof depends on nontrivial algorithms from computer
+ algebra and increase the reasoning power of the theorem prover. The
+ unsoundness of computer algebra systems is a major problem in
+ interfacing them to theorem provers. Our approach to obtaining a sound
+ overall system is not blanket distrust but based on the distinction
+ between algorithms we call sound and {\sl ad hoc} respectively. This
+ distinction is blurred in most computer algebra systems OUr
+ experimental interface therefore uses a computer algebra library. It
+ is based on theorem templates, which provide formal specifications for
+ the algorithms."
+}
+
+\end{chunk}
+
+\index{Beeson, Michael}
+\begin{chunk}{axiom.bib}
+@misc{Beesxx,
+ author = "Beeson, Michael",
+ title = "Automatic Generation of EpsilonDelta Proofs of Continuity",
+ url = "http://www.michaelbeeson.com/research/papers/aisc.pdf",
+ paper = "Beesxx.pdf",
+ abstract =
+ "As part of a project on automatic generation of proofs involving both
+ logic and computation, we have automated the production of some proofs
+ involving epsilondelta arguments. These proofs involve two or three
+ quantifiers on the logical side, and on the computational side, they
+ involve algebra, trigonometry, and some calculus. At the border of
+ logic and computation, they involve several types of arguments
+ involving inequalities, including transitivity chaining and several
+ types of bounding arguments, in which bounds are sought that do not
+ depend on certain variables. Control mechanisms have been developed
+ for intermixing logical deduction steps with computational steps and
+ with inequality reasoning. Problems discussed here as examples involve
+ the continuity and uniform continuity of various specific functions."
+}
+
+\end{chunk}
+
\index{Bressoud, David}
\begin{chunk}{axiom.bib}
@article{Bres93,
@@ 4145,13 +4204,14 @@ Martin, U.
\index{Poll, Erik}
\index{Thompson, Simon}
\begin{chunk}{axiom.bib}
@misc{Pollxx,
+@misc{Poll98,
author = "Poll, Erik and Thompson, Simon",
title = "Adding the axioms to Axiom. Toward a system of automated
reasoning in Aldor",
+ year = "1998",
url =
"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.1457&rep=rep1&type=ps",
 paper = "Pollxx.pdf",
+ paper = "Poll98.pdf",
keywords = "axiomref",
abstract = "
This paper examines the proposal of using the type system of Axiom to
@@ 4458,6 +4518,43 @@ Martin, U.
\end{chunk}
+\index{Schwarzweller, Christoph}
+\begin{chunk}{axiom.bib}
+@phdthesis{Schw97,
+ author = "Schwarzweller, Christoph",
+ title = "MIZAR verification of generic algebraic algorithms",
+ school = "University of Tubingen",
+ year = "1997",
+ paper = "Schw97.pdf",
+ keywords = "axiomref",
+ abstract =
+ "Although generic programming founds more and more attention –
+ nowadays generic programming languages as well as generic libraries
+ exist – there are hardly approaches for the verification of generic
+ algorithms or generic libraries. This thesis deals with generic
+ algorithms in the field of computer algebra. We propose the Mizar
+ system as a theorem prover capable of verifying generic algorithms on
+ an appropriate abstract level. The main advantage of the MIZAR theorem
+ prover is its special input language that enables textbook style
+ presentation of proofs. For generic versions of Brown/Henrici addition
+ and of Euclidean’s algorithm we give complete correctness proofs
+ written in the MIZAR language.
+
+ Moreover, we do not only prove algorithms correct in the usual
+ sense. In addition we show how to check, using the MIZAR system, that
+ a generic algebraic algorithm is correctly instantiated with a
+ particular domain. Answering this question that especially arises if
+ one wants to implement generic programming languages, in the field of
+ computer algebra requires nontrivial mathematical knowledge.
+
+ To build a verification system using the MIZAR theorem prover, we also
+ implemented a generator which almost automatically computes for a
+ given algorithm a set of theorems that imply the correctness of this
+ algorithm."
+}
+
+\end{chunk}
+
\index{Th\'ery, Laurent}
\begin{chunk}{axiom.bib}
@article{Ther01,
@@ 4960,9 +5057,11 @@ FME'99, Toulouse, France, Sept 2024, 1999, pp 17581777
\index{Martin, Ursula}
\index{Shand, D.}
\begin{chunk}{ignore}
\bibitem[Martin 97]{Mart97} Martin, U.; Shand, D.
 title = "Investigating some Embedded Verification Techniques for Computer Algebra Systems",
+\begin{chunk}{axiom.bib}
+@misc{Mart97,
+ author = "Martin, Ursula and Shand, D",
+ title = "Investigating some Embedded Verification Techniques for
+ Computer Algebra Systems",
url = "http://www.risc.jku.at/conferences/Theorema/papers/shand.ps.gz",
paper = "Mart97.ps",
abstract = "
@@ 4974,6 +5073,7 @@ FME'99, Toulouse, France, Sept 2024, 1999, pp 17581777
automated theorem provers and discuss possible advantages and
disadvantages of these approaches. We also discuss some possible case
studies."
+}
\end{chunk}
@@ 9930,8 +10030,8 @@ J. Symbolic Computation 5, 237259 (1988)
paper = "Riob92.pdf",
keywords = "axiomref",
comment = "\newline\refto{domain RECLOS RealClosure}",
 abstract = "
 Real algebraic numbers appear in many Computer Algebra problems. For
+ abstract =
+ "Real algebraic numbers appear in many Computer Algebra problems. For
instance the determination of a cylindrical algebraic decomposition
for an euclidean space requires computing with real algebraic numbers.
This paper describes an implementation for computations with the real
@@ 9943,8 +10043,216 @@ J. Symbolic Computation 5, 237259 (1988)
\end{chunk}
+\section{Comparison of Computer Algebra System} %%%%%%%%%%%%%%%%%%%%%%
+
+\index{Bernardin, Laurent}
+\begin{chunk}{axiom.bib}
+@article{Bern96,
+ author = "Benardin, Laurent",
+ title = "A review of symbolic solvers",
+ journal = "SIGSAM Bull.",
+ volume = "30",
+ number = "1",
+ pages = "920",
+ year = "1996",
+ keywords = "axiomref",
+ paper = "Bern96.pdf",
+ abstract =
+ "Solving equations and systems of equations symbolically is a key
+ feature of every Computer Algebra System. This review examines the
+ capabilities of the six best known general purpose systems to date in
+ the area of general algebraic and transcendental equation
+ solving. Areas explicitly not covered by this review are differential
+ equations and numeric or polynomial system solving as special purpose
+ systems exist for these kinds of problems. The aim is to provide a
+ benchmark for comparing Computer Algebra Systems in a specific
+ domain. We do not intend to give a rating of overall capabilities as
+ for example in [9]. 1 The Contestants We compare six major Computer
+ Algebra Systems. Axiom 2.0 [7], Derive 3.06 [1], Macsyma 420 [8],
+ Maple V R4 [3], Mathematica 2.2 [10], MuPAD 1.2.9 [5] and Reduce 3.6
+ [6]. When available, we tried to use the latest shipping version of
+ each system. 2 The Problem Set The following table presents the set of
+ 80 problems that we used to evaluate the different solvers..."
+}
+
+\end{chunk}
+
+\index{Wester, Michael J.}
+\begin{chunk}{axiom.bib}
+@misc{West95,
+ author = "Wester, Michael J.",
+ title = "A Review of CAS Mathematical Capabilities",
+ year = "1995",
+ keywords = "axiomref",
+ paper = "West95.pdf",
+ url = "http://math.unm.edu/~wester/cas/Paper.ps",
+ abstract =
+ "Computer algebra systems (CASs) have become an important
+ computational tool in the last decade. General purpose CASs, which are
+ designed to solve a wide variety of problems, have gained special
+ prominance. In this paper, the capabilities of seven major general
+ purpose CASs (Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, and
+ Reduce) are reviewed on 131 short problems covering a broad range of
+ (primarily) symbolic mathematics.
+
+ A demo was developed for each CAS, run and the results
+ evaluated. Problems were graded in terms of whether it was easy or
+ difficult or possible to produce an answer and if an answer was
+ produced, whether it was correct. It is the author's hope that this
+ review will encourage the development of a comprehensive CAS test
+ suite."
+}
+
+\end{chunk}
+
+\index{Wester, Michael J.}
+\begin{chunk}{axiom.bib}
+@misc{West99a,
+ author = "Wester, Michael J.",
+ title = "A Critique of the Mathematical Abilities of CA Systems",
+ year = "1999",
+ url = "http://math.unm.edu/~wester/cas/book/Wester.pdf",
+ url2 = "http://math.unm.edu/~wester/cas_review.html",
+ paper = "West99a.pdf",
+ abstract =
+ "Computer algebra systems (CASs) have become an essential computational
+ tool in the last decade. General purpose CASs, which are designed to
+ solve a wide variety of problems, have gained special prominence. In
+ this chapter, the capabilities of seven major general purpose CASs
+ (Axiom, Derive, Macsyma, Maple, Mathmatica, MuPAD and Reduce) are
+ reviewed on 542 short problems covering a broad range of (primarily)
+ symbolic mathematics."
+
+}
+
+\end{chunk}
+
\section{To Be Classified} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\index{Aslaksen, Helmer}
+\begin{chunk}{axiom.bib}
+@article{Asla96,
+ author = "Aslaksen, Helmer",
+ title = "Multiplevalued complex functions and computer algebra",
+ journal = "SIGSAM Bulletin",
+ volume = "30",
+ number = "2",
+ year = "1996",
+ pages = "1220",
+ paper = "Asla96.pdf",
+ url = "http://www.math.nus.edu.sg/aslaksen/papers/cacas.pdf",
+ abstract =
+ "I recently taught a course on complex analysis. That forced me to
+ think more carefully about branches. Being interested in computer
+ algebra, it was only natural that I wanted to see how such programs
+ dealt with these problems. I was also inspired by a paper by
+ Stoutemyer.
+
+ While programs like Derive, Maple, Mathematica and Reduce are very
+ powerful, they also have their fair share of problems. In particular,
+ branches are somewhat of an Achilles' heel for them. As is wellknown,
+ the complex logarithm function is properly defined as a
+ multiplevalued function. And since the general power and exponential
+ functions are defined in terms of the logarithm function, they are
+ also multiplevalued. But for actual computations, we need to make
+ them single valued, which we do by choosing a branch. In Section 2, we
+ will consider some transformation rules for branches of
+ multiplevalued complex functions in painstaking detail.
+
+ The purpose of this short article is not to do a comprehensive
+ comparative study of different computer algebra systems. My goal is
+ simply to make the readers aware of some of the problems, and to
+ encourage the readers to sit down and experiment with their favourite
+ programs."
+}
+
+\end{chunk}
+
+\index{Tonisson, Eno}
+\begin{chunk}{axiom.bib}
+@article{Tonixx,
+ author = "Tonisson, Eno",
+ title = "Branch Completeness in School Mathematics and in Computer Algebra
+ Systems",
+ journal = "The Electronic Journal of Mathematics and Technology",
+ volume = "1",
+ number = "1",
+ issn = "19332823",
+ paper = "Tonixx.pdf",
+ url = "https://php.radford.edu/~ejmt/deliveryBoy.php?paper=eJMT_v1n3p5",
+ abstract =
+ "In many cases when solving school algebra problems (e.g. simplifying
+ an expression, solving an equation), the solution is separable into
+ branches in some manner. The paper describes some approaches to
+ branches that are used in school textbooks and computer algebra
+ systems and compares them with mathematically branchcomplete
+ solutions. It tries to identify possible reasons behind different
+ approaches and also indicate some ideas how such differences could be
+ explained to the students."
+}
+
+\end{chunk}
+
+\index{Stoutemyer, David R.}
+\begin{chunk}{axiom.bib}
+@article{Stou91,
+ author = "Stoutemyer, David R.",
+ title = "Crimes and misdemeanors in the computer algebra trade",
+ journal = "Notices of the American Mathematical Society",
+ volume = "38",
+ number = "7",
+ pages = "778785",
+ year = "1991"
+}
+
+\end{chunk}
+
+\index{Sangwin, Chris}
+\begin{chunk}{axiom.bib}
+@misc{Sang10,
+ author = "Sangwin, Chris",
+ title = "Intriguing Integrals: Part I and II",
+ year = "2010",
+ url1 =
+ "https://plus.maths.org/issue54/features/sangwin/2pdf/index.html/op.pdf",
+ paper1 = "Sang10a.pdf",
+ url2 =
+ "https://plus.maths.org/issue54/features/sangwin2/2pdf/index.html/op.pdf",
+ paper2 = "Sang10b.pdf"
+}
+
+\end{chunk}
+
+\index{Evans, Brian}
+\begin{chunk}{axiom.bib}
+@misc{Evanxx,
+ author = "Evans, Brian",
+ title = "History of CA Systems",
+ url = "http://felix.unife.it/Root/dMathematics/dThemathematician/dHistoryofmathematics/tHistoryofcomputeralgebra",
+ paper = "Evanxx.txt"
+}
+
+\end{chunk}
+
+\index{Apel, Joachim}
+\index{Klaus, Uwe}
+\begin{chunk}{axiom.bib}
+@misc{Apel94,
+ author = "Apel, Joachim and Klaus, Uwe",
+ title = "Representing Polynomials in Computer Algebra Systems",
+ year = "1994",
+ paper = "Apel94.pdf",
+ abstract =
+ "There are discussed implementational aspects of the specialpurpose
+ computer algebra system FELIX designed for computations in
+ constructive algebra. In particular, data types developed for the
+ representation of and computation with commutative and noncommuative
+ polynomials are described. Furthermore, comparison of time and memory
+ requirements of different polynomial representations are reported."
+}
+
+\end{chunk}
+
\index{Judson, Thomas W.}
\begin{chunk}{axiom.bib}
@book{Juds15,
@@ 10701,6 +11009,35 @@ J. Symbolic Computation 5, 237259 (1988)
\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\index{Ablamowicz, Rafal}
+\index{Lounesto, Pertti}
+\index{Para, Josep M.}
+\begin{chunk}{axiom.bib}
+@book{Abla96,
+ author = "Ablamowicz, Rafal and Lounesto, Pertti and Para, Josep M.",
+ title = "Clifford algebras with numeric and symbolic computations",
+ publisher = "Birkhauser",
+ year = "1996",
+ keyword = "axiomref",
+ abstract =
+ "The 20 articles of this volume will be reviewed individually. They
+ have been grouped into the following four sections: I) Verifying and
+ falsifying conjectures (1); II) Differential geometry quantum
+ mechanics, spinors and conformal group (9); III) Generalized Clifford
+ algebras and number systems, projective geometry and crystallography
+ (7); IV) Numerical methods in Clifford algebras (3).
+
+ Selected electronic materials submitted by our contributors can be
+ found at a Web site:
+ http://www.birkhauser.com/books/ISBN/0817639071
+
+ These materials contain original packages, worksheets, notebooks,
+ computer programs, etc., that were used in deriving results presented
+ in this book."
+}
+
+\end{chunk}
+
\index{Kokolvoljc, Vlasta}
\index{Kutzler, Bernhard}
\begin{chunk}{axiom.bib}
@@ 12097,7 +12434,10 @@ Informatique et en Automatique, Le Chesnay, France, 12pp
title = "Utilisation de logiciels libres pour la r\'ealisation de TP MT26",
year = "2004",
paper = "Carp04.pdf",
 keywords = "axiomref"
+ url = "http://axiomwiki.newsynthesis.org/public/refs/ac20.pdf",
+ keywords = "axiomref",
+ comment = "french",
+ abstract = "radicalSolve(x**3+x**27=0,x)"
}
\end{chunk}
@@ 12159,6 +12499,84 @@ Proc. Natl. Acad. Sci. USA Vol 86
\end{chunk}
+\index{Colin, Antoine}
+\begin{chunk}{axiom.bib}
+@article{Coli97,
+ author = "Colin, Antoine",
+ title = "Solving a system of algebraic equations with symmetries",
+ journal = "J. Pure Appl. Algebra",
+ volume = "117118",
+ pages = "195215",
+ year = "1997",
+ keywords = "axiomref",
+ abstract =
+ "Let $(F)$ be a system of $p$ polynomial equations
+ $F_i({\bf X}) \in k[{\bf X}]$, where $k$ is a commutative field and
+ ${\bf X} := (X_1,\cdots,X_n)$ are indeterminates. Let $G$ be a subgroup
+ of $GL_n(k)$. A polynomial $P \in k[{\bf X}]$ (resp. rational function
+ $P \in k({\bf X})$ ) is an invariant of $G$ if and only if for all
+ $A \in G$ we have $A\cdot P = P$. We denote $k[{\bf X}]^G$ by (resp.
+ $k({\bf X})^G$) the algebra of polynomial (resp. rational function)
+ invariants of $G$. If $L$ is another subgroup of $GL_n(k)$ such that
+ $G \subset L$, $P$ is called a primary invariant of $G$ relative to $L$ if
+ and only if $Stab_L(P) = G$ (where $Stab_L(P)$ is the stabilizer of
+ $P$ in $L$).
+
+ The paper describes the algebra of the invariants of a finite group
+ and how to express these invariants in terms of a small number of
+ them, from both the CohenMacaulay algebra and the field theory points
+ of view. A method is proposed to solve $(F)$ by expressing it in terms of
+ primary invariants $\Pi_1,\cdots,\Pi_n$
+ (e.g. the elementary symmetric polynomials) and one
+ ``primitive'' secondary invariant.
+
+ The main thrust of the paper is contained in the following theorem.
+ Let $(F)$ be a set of invariants of $G$. Let $L$ be a subgroup of
+ $GL_n(k)$ such that $G \subset L$ and $k({\bf X})^L$ is a purely
+ transcendental extension of $k_i$, let $\Pi_1,\cdots,\Pi_n$ be
+ polynomials such that $k({\bf X})^L = k(\Pi_1,\cdots,\Pi_n)$,
+ and let $\Theta \in k[{\bf X}]^G$ be a primitive polynomial invariant
+ of $G$ relative to $L$.
+ When possible, it is convenient to choose $\Theta$ to be one of the
+ polynomials in $(F)$. – An algorithm is given that allows each polynomial
+ $F_i$ to be expressed as $F_i({\bf X}) = H_i(\Pi_1,\cdots,\Pi_n,\Theta)$,
+ an algebraic fraction in $\Pi_1,\cdots,\Pi_n$ and a polynomial in
+ $\Theta$. Now let $L$ be the minimal polynomial of $\Theta$ over
+ $k[{\bf X}]^L$; we have
+ \[L({\bf X},T)=\prod_{\Theta^{'} \in L\cdot \Theta}(T\Theta^{'})
+ \in k[{\bf X}]^L[T]\]
+ (where $L$ is called a generic Lagrange resolvent).
+ As $k(\Pi_1,\cdots,\Pi_n)=k({\bf X})^L$, we can write
+ $L({\bf X},T)=H_0(\Pi_1,\cdots,\Pi_n,T)$ where $H_0$ is some
+ rational function. The question
+ $H_0(\Pi_1,\cdots,\Pi_n,\Theta)=0$ is always satisfied because
+ $\Theta$ is a root of $L$. Then, we solve the system of ($p=1$)
+ algebraic equations $H_i(\Pi_1,\cdots,\Pi_n,\Theta)=0$,
+ $0 \le i \le p$ for $\Pi_1,\cdots,\Pi_n,\Theta$ as indeterminates.
+
+ Theorem 1: Let $D \in k[\Pi_1,\cdots,\Pi_n]$ be the LCM of the
+ denominators of all the fractions $H_i$,$0 \le i \le p$ and let
+ $H_i^{'}=DH_i$. For every solution
+ $x:=(x_1,\cdots,x_n)$ of the system $(F)$:$F_i({\bf X})=0$,
+ $1 \le i \le p$, there exists a solution ($\pi_1,\cdots,\pi_n,\Theta$)
+ of the system
+ $(H^{'}):H_i^{'}(\Pi_1,\cdots,\Pi_n,\Theta)=0$, $0 \le i \le p$ such
+ that $x$ is a solution of the system
+ $(P_\pi):\Pi_i({\bf X})=\pi_i$, $1 \le i \le n$ , and of the equation
+ $\Theta({\bf X})=0$. Conversely, for any solution
+ $(\pi_1,\cdots,\pi_n,\theta)$ of the system $(H^{'})$ such that
+ $D(\pi_1,\cdots,\pi_n) \ne 0$, if $x$ is a solution of the system
+ $(P_\pi)$ relative to $(\pi_1,\cdots,\pi_n)$, then there exists
+ some $A \in L$ such that $\Theta(A\cdot x)=\theta$, and then for all
+ $B \in G$, $BA\cdot x$, is a solution of the system $(F)$.
+
+ A slighly more general version of this theorem is also given. The
+ paper then presents an algorithm that applies the theory and has been
+ implemented in AXIOM. It is followed by several examples."
+}
+
+\end{chunk}
+
\index{Collins, G.E.}
\index{Mignotte, M.}
\index{Winkler, F.}
@@ 12248,13 +12666,14 @@ Coding Theory and Applications Proceedings. SpringerVerlag, Berlin, Germany
\index{French, Tim}
\index{Maple, Carsten}
\index{Pott, Sandra}
\begin{chunk}{ignore}
\bibitem[Conrad (a)]{CFMPxxa}
+\begin{chunk}{axiom.bib}
+@misc{Conrxxa,
author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra",
title = "Approaching Inheritance from a Natural Mathematical Perspective
and from a Java Driven Viewpoint: a Comparative Review",
keywords = "axiomref",
 paper = "CFMPxxa.pdf",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/McTfCmSpaxiom.pdf",
+ paper = "Conrxxa.pdf",
abstract = "
It is wellknown that few objectoriented programming languages allow
objects to change their nature at runtime. There have been a number
@@ 12265,6 +12684,7 @@ Coding Theory and Applications Proceedings. SpringerVerlag, Berlin, Germany
from this context that we present a framework that realistically
represents the dynamic and evolving characteristic of problems and
algorithms."
+}
\end{chunk}
@@ 12514,11 +12934,16 @@ Mathematical Sciences Department, IBM Thomas Watson Research Center 1984
\end{chunk}
\index{Davenport, James H.}
\begin{chunk}{ignore}
\bibitem[Davenport 84a]{Dav84a} Davenport, James H.
+\begin{chunk}{axiom.bib}
+@misc{Dave84a,
+ author = "Davenport, James H.",
title = "A New Algebra System",
 paper = "Dav84a.pdf",
+ paper = "Dave84a.pdf",
keywords = "axiomref",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/Davenport1984a\_new\_algebra\_system.pdf",
+ abstract =
+ "Seminal internal paper discussing Axiom design decisions."
+}
\end{chunk}
@@ 12748,21 +13173,24 @@ Oxford, UK, December 1992
\end{chunk}
\index{Davenport, James H.}
\begin{chunk}{ignore}
\bibitem[Davenport 92b]{Dav92b} Davenport, J. H.
+\begin{chunk}{axiom.bib}
+@techreport{Dave92b,
+ author = "Davenport, James H.",
title = "How does one program in the AXIOM system?",
AXIOM Technical Report TR6/92 (ATR/4)(NP2493)
Numerical Algorithms Group, Inc., Downer's
Grove, IL, USA and Oxford, UK December 1992
+ institution = "Numerical Algorithms Group, Inc.",
+ year = "1992",
+ type = "technical report",
+ number = "TR6/92 (ATR/4)(NP2493)",
url = "http://www.nag.co.uk/doc/TechRep/axiomtr.html",
 paper = "Dav92b.pdf",
+ paper = "Dave92b.pdf",
keywords = "axiomref",
 abstract = "
 Axiom is a computer algebra system superficially like many others, but
+ abstract =
+ "Axiom is a computer algebra system superficially like many others, but
fundamentally different in its internal construction, and therefore in
the possibilities it offers to its users and programmers. In these
lecture notes, we will explain, by example, the methodology that the
author uses for programming substantial bits of mathematics in Axiom."
+}
\end{chunk}
@@ 12801,11 +13229,13 @@ December 1992.
\index{Davenport, James H.}
\index{Faure, Christ\'ele}
\begin{chunk}{ignore}
\bibitem[Davenport (a)]{DFxx} Davenport, James; Faure, Christ\'ele
+\begin{chunk}{axiom.bib}
+@misc{Davexx,
+ author = {Davenport, James; Faure, Christ\'ele},
title = "The Unknown in Computer Algebra",
 url = "http://axiomwiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf",
 paper = "DFxx.pdf",
+ url =
+"http://axiomwiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf",
+ paper = "Davexx.pdf",
keywords = "axiomref",
abstract = "
Computer algebra systems have to deal with the confusion between
@@ 12814,6 +13244,7 @@ December 1992.
are unknown, but whose type is known. For examples $x^p \ne x$ if $x$
is a symbol, but $x^p = x$ if $x \in GF(p)$. We show how we have
extended Axiom to deal with this concept."
+}
\end{chunk}
@@ 12902,6 +13333,64 @@ and Laine, M. and Valkeila, E. pp112 University of Helsinki, Finland (1994)
\end{chunk}
+\index{DiBlasio, Paolo}
+\index{Temperini, Marco}
+\begin{chunk}{axiom.bib}
+@article{DiBl95,
+ author = "DiBlasio, Paolo and Temperini, Marco",
+ title = "Subtyping Inheritance and Its Application in Languages for
+ Symbolic Computation Systems",
+ journal = "J. Symbolic Computation",
+ volume = "19",
+ pages = "3963",
+ year = "1995",
+ paper = "DiBl95.pdf",
+ keywords = "axiomref",
+ abstract =
+ "Application of objectoriented programming techniques to design and
+ implementation of symbolic computation is investigated. We show the
+ significance of certain correctness problems, occurring in programming
+ environments based on specialization inheritance, due to use of method
+ redefinition and polymorphism. We propose a solution to these
+ problems, by defining a mechanism of subtyping inheritance and the
+ prototype of an objectoriented programming language for a symbolic
+ computation system. We devise the subtyping inheritance {\sl ESI
+ (Enhanced String Inheritance)} by lifting to programming language
+ constructs a given model of subtyping, which is established by a
+ monotonic (covariant) subtyping rule. Type safeness of language
+ instructions is proved.
+
+ The adoption of {\sl ESI} allows to model method and class
+ specialization in a natural way. The {\sl ESI} mechanism verifies the
+ type correctness of language statements by means of type checking
+ rules and preserves their correctness at runtime by a suitable method
+ lookup algorithm."
+}
+
+\end{chunk}
+
+\index{DiBlasio, Paolo}
+\index{Temperini, Marco}
+\begin{chunk}{axiom.bib}
+@InProceedings{DiBl97,
+ author = "DiBlasio, Paolo and Temperini, Marco",
+ title = "On subtyping in languages for symbolic computation systems",
+ booktitle = "Advances in the design of symbolic computation systems",
+ series = "Monographs in Symbolic Computation",
+ year = "1997",
+ publisher = "Springer",
+ pages = "164178",
+ keywords = "axiomref",
+ abstract =
+ "We want to define a strongly typed OOP language suitable as the
+ software development tool of a symbolic computation system, which
+ provides class structure to manage ADTs and supports multiple
+ inheritance to model specialization hierarchies. In this paper, we
+ provide the theoretical background for such a task."
+}
+
+\end{chunk}
+
\index{Dicrescenzo, C.}
\index{Duval, Dominique}
\begin{chunk}{ignore}
@@ 13230,6 +13719,37 @@ Madrid Spain, NAG conference (private copy of paper)
\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\index{Fakler, Winfried}
+\begin{chunk}{axiom.bib}
+@article{Fakl97,
+ author = "Fakler, Winfried",
+ title = "On second order homogeneous linear differential equations with
+ Liouvillian solutions",
+ journal = "Theor. Comput. Sci.",
+ volume = "187",
+ number = "12",
+ pages = "2748",
+ year = "1997",
+ paper = "Fakl97.pdf",
+ keywords = "axiomref",
+ abstract =
+ "We determine all minimal polynomials for second order homogeneous
+ linear differential equations with algebraic solutions decomposed into
+ invariants and we show how easily one can recover the known conditions
+ on differential Galois groups [J. Kovacic, J. Symb. Comput. 2, 343
+ (1986; Zbl 0603.68035), M. F. Singer and F. Ulmer,
+ J. Symb. Comput. 16, 936, 3773 (1993; Zbl 0802.12004, Zbl
+ 0802.12005), F.Ulmer and J. A. Weil, J. Symb. Comput. 22, 179200
+ (1996; Zbl 0871.12008)] using invariant theory. Applying these
+ conditions and the differential invariants of a differential equation
+ we deduce an alternative method to the algorithms given in (loc. cit.)
+ for computing Liouvillian solutions. For irreducible second order
+ equations our method determines solutions by formulas in all but three
+ cases."
+}
+
+\end{chunk}
+
\index{Farmer, William M.}
\index{von Mohrenschildt, Martin}
\begin{chunk}{axiom.bib}
@@ 13276,6 +13796,31 @@ In Watanabe and Nagata [WN90], pp6067 ISBN 0897914015 LCCN QA76.95.I57 1990
\index{Fateman, Richard J.}
\begin{chunk}{axiom.bib}
+@InProceedings{Fate96,
+ author = "Fateman, Richard J.",
+ title = "A Review of Symbolic Solvers",
+ booktitle = "Proc 1996 ISSAC",
+ series = "ISSAC 96",
+ year = "1996",
+ pages = "8694",
+ keywords = "axiomref",
+ keywords = "axiomref",
+ paper = "Fate96.pdf",
+ url = "http://http.cs.berkeley.edu/~fateman/papers/eval.ps",
+ abstract =
+ "``Evaluation'' of expressions and programs in a computer algebra
+ system is central to every system, but inevitably fails to provide
+ complete satisfaction. Here we explain the conflicting requirements,
+ describe some solutions from current systems, and propose alternatives
+ that might be preferable sometimes. We give examples primarily from
+ Axiom, Macsyma, Maple, Mathematica, with passing metion of a few other
+ systems."
+}
+
+\end{chunk}
+
+\index{Fateman, Richard J.}
+\begin{chunk}{axiom.bib}
@InProceedings{Fate00,
author = "Fateman, Richard J.",
title = "Problem solving environments and symbolic computing",
@@ 13307,13 +13852,20 @@ In Watanabe and Nagata [WN90], pp6067 ISBN 0897914015 LCCN QA76.95.I57 1990
\end{chunk}
\index{Fateman, Richard J.}
\begin{chunk}{ignore}
\bibitem[Fateman 05]{Fat05} Fateman, R. J.
 title = "An incremental approach to building a mathematical expert out of software",
4/19/2005\hfill
+\begin{chunk}{axiom.bib}
+@misc{Fate05,
+ author = "Fateman, Richard J.",
+ title = "An incremental approach to building a mathematical
+ expert out of software",
+ conference = "Axiom Computer Algebra Conference",
+ location = "City College of New York, CAISS project",
+ year = "2005",
+ month = "April",
+ day = "19",
url = "http://www.cs.berkeley.edu/~fateman/papers/axiom.pdf",
paper = "Fat05.pdf",
 keywords = "axiomref",
+ keywords = "axiomref"
+}
\end{chunk}
@@ 13732,18 +14284,93 @@ In Fitch [Fit93], pp193202. ISBN 0387572724 (New York),
\end{chunk}
\index{Gr\"abe, HansGert}
\begin{chunk}{ignore}
\bibitem[Grabe 98]{Gra98} Gr\"abe, HansGert
 title = "About the Polynomial System Solve Facility of Axiom, Macyma, Maple Mathematica, MuPAD, and Reduce",
 paper = "Gra98.pdf",
+\begin{chunk}{axiom.bib}
+@misc{Grab98,
+ author = "Grabe, HansGert",
+ title = "About the Polynomial System Solve Facility of Axiom, Macsyma,
+ Maple Mathematica, MuPAD, and Reduce",
+ paper = "Grab98.pdf",
+ url =
+"https://www.informatik.unileipzig.de/~graebe/ComputerAlgebra/Publications/WesterBook.pdf",
keywords = "axiomref",
 abstract = "
 We report on some experiences with the general purpose Computer
+ abstract =
+ "We report on some experiences with the general purpose Computer
Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and
Reduce solving systems of polynomial equations and the way they
present their solutions. This snapshot (taken in the spring of 1996)
of the current power of the different systems in a special area
concentrates on both CPUtimes and the quality of the output."
+}
+
+\end{chunk}
+
+\index{Gr\"abe, HansGert}
+\begin{chunk}{axiom.bib}
+@misc{Grab06,
+ author = "Grabe, HansGert",
+ title = "The Groebner Factorizer and Polynomial System Solving",
+ year = "2006",
+ keywords = "axiomref",
+ report = "Special Semester on Groebner Bases",
+ location = "Linz",
+ paper = "Grab06.pdf",
+ url =
+"https://www.ricam.oeaw.ac.at/specsem/srs/groeb/download/06\_02\_Solver.pdf",
+ abstract =
+ "Let $S := k[x_1,\ldots, x_n]$ be the polynomial ring in the
+ variables $x_1,\ldots,x_n$ over the field $k$ and
+ $B := \{f_1,\ldots,f_m\} \subset S$
+ be a finite system of polynomials. Denote by $I(B)$ the
+ ideal generated by these polynomials. One of the major tasks of
+ constructive commutative algebra is the derivation of information
+ about the structure of
+ \[V(B):=\{a \in K^n : \forall f \in B{\rm\ such\ that\ }f(a)=0\}\]
+ the set of common zeroes of the system $B$ over an
+ algebraically closed extension $K$ of $k$. Splitting the system into
+ smaller ones, solving them separately, and patching all solutions
+ together is often a good guess for a quick solution of even highly
+ nontrivial problems. This can be done by several techniques, e.g.,
+ characteristic sets, resultants, the Groebner factorizer or some ad
+ hoc methods. Of course, such a strategy makes sense only for problems
+ that really will split, i.e., for reducible varieties of
+ solutions. Surprisingly often, problems coming from 11real life''
+ fulfill this condition.
+
+ Among the methods to split polynomial systems into smaller pieces
+ probably the Groebner factor izer method attracted the most
+ theoretical attention, see Czapor ([4, 5]), Davenport ([6]), Melenk, M
+ ̈oller and Neun ([16, 17]) and Gr ̈abe ([13, 14]). General purpose
+ Computer Algebra Systems (CAS) are well suited for such an approach,
+ since they make available both a (more or less) well tuned
+ implementation of the classical Groebner algorithm and an effective
+ multivariate polynomial factorizer.
+
+ Furthermore it turned out that the Groebner factorizer is not only a
+ good heuristic approach for splitting, but its output is also usually
+ a collection of almost prime components. Their description allows a
+ much deeper understanding of the structure of the set of zeroes
+ compared to the result of a sole Groebner basis computation.
+
+ Of course, for special purposes a general CAS as a multipurpose
+ mathematical assistant can’t offer the same power as specialized
+ software with efficiently implemented and well adapted algorithms and
+ data types. For polynomial system solving, such specialized software
+ has to implement two algorithmically complex tasks, solving and
+ splitting, and until recently none of the specialized systems (as
+ e.g., GB, Macaulay, Singular, CoCoA, etc.) did both
+ efficiently. Meanwhile, being very efficient computing (classical)
+ Groebner bases, development efforts are also directed, not only
+ for performance reasons, towards a better inclusion of factorization
+ into such specialized systems. Needless to remark that it needs some
+ skill to force a special system to answer questions and the user will
+ probably first try his ``home system'' for an answer. Thus the
+ polynomial systems solving facility of the different CAS should behave
+ especially well on such polynomial systems that are hard enough not to
+ be done by hand, but not really hard to require special efforts. It
+ should invoke a convenient interface to get the solutions in a form
+ that is (correct and) well suited for further analysis in the familiar
+ environment of the given CAS as the personal mathematical assistant."
+}
\end{chunk}
@@ 13788,15 +14415,17 @@ SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
\index{Griesmer, James H.}
\index{Jenks, Richard D.}
\begin{chunk}{ignore}
\bibitem[Griesmer 71]{GJ71} Griesmer, J. H.; Jenks, R.D.
+\begin{chunk}{axiom.bib}
+@InProceedings{Grie71,
+ author = "Griesmer, James H. and Jenks, Richard D.",
title = "SCRATCHPAD/1  an interactive facility for symbolic mathematics",
In Petrick [Pet71], pp4258. LCCN QA76.5.S94 1971
+ booktitle = "Proc. second ACM Symposium on Symbolic and Algebraic
+ Manipulation",
+ series = "SYMSAC 71",
+ year = "1971",
+ pages = "4258",
url = "http://delivery.acm.org/10.1145/810000/806266/p42griesmer.pdf",
SYMSAC'71 Proc. second ACM Symposium on Symbolic and Algebraic
Manipulation pp4548
paper = "GJ71.pdf",
 ref = "00027",
keywords = "axiomref",
abstract = "
The SCRATCHPAD/1 system is designed to provide an interactive symbolic
@@ 13806,6 +14435,7 @@ Manipulation pp4548
introducing new notations into the language. A comprehensive system
library incorporates symbolic capabilities provided by such systems as
SIN, MATHLAB, and REDUCE."
+}
\end{chunk}
@@ 14307,11 +14937,49 @@ Developments. LIFL Univ. Lille, Lille France, 1993
\end{chunk}
\index{Jacquemard, A.}
+\index{Jacquemard, Alain}
+\index{KhechichineMourtada, F.Z.}
+\index{Mourtada, A.}
+\begin{chunk}{axiom.bib}
+@article{Jacq97,
+ author = "Jacquemard, Alain and KhechichineMourtada, F.Z. and Mourtada, A.",
+ title = "Formal algorithms applied to the study of the cyclicity of a
+ generic algebraic polycycle with four hyperbolic crests",
+ journal = "Nonlinearity",
+ volume = "10",
+ number = "1",
+ pages = "1953",
+ year = "1997",
+ keywords = "axiomref",
+ comment = "french",
+ abstract =
+ "Drawing on the work of Mourtada, we show that a family of vector
+ fields with a generic algebraic polycycle of four hyperbolic apices
+ possesses a maximum capacity of four limit cycles. This cyclicity is
+ attained in an opening connecting the parameters which the edge
+ contains, in particular a generic line of singularities of dovetail
+ type. We also give an asymptotic estimation of the volume of this
+ opening, as well as an explicit example of a family of polynomial
+ vector fields replicating the abovedescribed conditions and
+ possessing five limit cycles. The methods employed are very diverse:
+ geometrical arguments (Thom’s theory of catastrophes and the theory of
+ algebraic singularities), developments from Puiseux, the number of
+ major roots by Descartes’ law and calculated exactly by Sturm series,
+ and other specific methods for formal calculus, such as for example
+ the cylindrical algebraic decomposition and the resolution of
+ algebraic systems via the construction of Gröbner bases. The
+ calculations have been executed formally, that is to say without
+ making the least appeal to numerical approximation, in using the
+ formal calculus system AXIOM."
+}
+
+\end{chunk}
+
+\index{Jacquemard, Alain}
\index{Teixeira, M.A.}
\begin{chunk}{axiom.bib}
@article{Jacq02,
 author = "Jacquemard, A. and Teixeira, M.A.",
+ author = "Jacquemard, Alain and Teixeira, M.A.",
title = "Effective algebraic geometry and normal forms of reversible
mappings",
journal = "Rev. Mat. Complut.",
@@ 15240,14 +15908,18 @@ CODEN JSYCEH ISSN 07477171
\index{Lambe, Larry A.}
\index{Luczak, Richard}
\begin{chunk}{ignore}
\bibitem[Lambe 93a]{LL93} Lambe, Larry; Luczak, Richard
 title = "ObjectOriented Mathematical Programming and Symbolic/Numeric Interface",
$3^{rd}$ International Conf. on Expert Systems in Numerical Computing 1993
 paper = "LL93.pdf",
+\begin{chunk}{axiom.bib}
+@article{Lamb93a,
+ article = "Lambe, Larry and Luczak, Richard",
+ title = "ObjectOriented Mathematical Programming and
+ Symbolic/Numeric Interface",
+ journal = "3rd Int. Conf. on Expert Systems in Numerical Computing",
+ year = "1993",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/axiomfem.pdf",
+ paper = "Lamb93a.pdf",
keywords = "axiomref",
 abstract = "
 The Axiom language is based on the notions of ``categories'',
+ abstract =
+ "The Axiom language is based on the notions of ``categories'',
``domains'', and ``packages''. These concepts are used to build an
interface between symbolic and numeric calculations. In particular, an
interface to the NAG Fortran Library and Axiom's algebra and graphics
@@ 15256,6 +15928,73 @@ $3^{rd}$ International Conf. on Expert Systems in Numerical Computing 1993
element method. While the examples are elementary, we believe that
they point to very powerful methods for combining numeric and symbolic
computational techniques."
+}
+
+\end{chunk}
+
+\index{Lambe, Larry A.}
+\index{Radford, David E.}
+\begin{chunk}{axiom.bib}
+@book{Lamb97,
+ author = "Lambe, Larry A. and Radford, David E.",
+ title = "Introduction to the quantum YangBaxter equation and quantum
+ groups: an algebraic approach",
+ booktitle = "Mathematics and its Applications",
+ publisher = "Kluwer Adademic Publishers",
+ year = "1997",
+ keywords = "axiomref",
+ abstract =
+ "The quantum YangBaxter equation (QYBE) has roots in statistical
+ mechanics and the inverse scattering method and leads to a natural
+ construction of a bialgebra. It turns out to have important
+ connections with knot theory and invariants of 3manifolds. There are
+ now available many reference books to quantum groups and these various
+ applications. The book under review develops the algebraic
+ underpinning and theory of the QYBE, including the constant form and
+ the one and two parameter forms.
+
+ We give a brief description of the chapters. Chapter 1 (together with
+ an Appendix) gives the algebraic preliminaries involving coalgebras,
+ bialgebras, Hopf algebras, modules and comodules. Chapter 2 introduces
+ the various forms of the QYBE, and the basic algebraic structures
+ associated to them, including FaddeevReshetikhinTakhtadzhan (FRT)
+ construction. Chapter 3 explores various categorical settings for the
+ constant form of the QYBE, the most basic being the category of left
+ QYB modules over a bialgebra and the notion of algebras, coalgebras,
+ etc. in this category. Chapter 4 develops universal mapping properties
+ of the FRT construction and its reduced version, and the authors
+ investigate when the reduced FRT construction leads to a pointed
+ bialgebra or a pointed Hopf algebra. Chapter 5 develops the quantum
+ groups associated to $SL(2)$, i.e., the quantum universal enveloping
+ algebra, and the quantum function algebra. Chapter 6 introduces
+ quasitriangular Hopf algebras, and discusses how the
+ finitedimensional ones give rise to solutions of the QYBE through
+ their representation theory. The most important example is the
+ Drinfeld double of a finitedimensional Hopf algebra. The authors note
+ (through an exercise!) that every finitedimensional Hopf algebra is
+ the reduced FRT construction of some solution to the QYBE. Chapter 7
+ introduces coquasitriangular bialgebras, the most important being the
+ FRT and the reduced FRT constructions. There are some generalizations
+ here to the oneparameter form of the QYBE. Chapter 8 uses all the
+ previously developed techniques to find solutions of the QYBE in
+ certain cases, including the oneparameter form. Some of these were
+ discovered by computer algebra methods. The final chapter 9 gives a
+ brief discussion of certain categorical constructions and the QYBE is
+ certain fairly abstract categories, motivated by the fact that the FRT
+ construction is a coend.
+
+ This book fills an important niche in the literature involving the
+ QYBE by highlighting the algebraic aspects and applications. Although
+ this is basically a reference book, it includes so many important
+ parts of the study of Hopf algebras that it could be used as a
+ textbook for a certain type of course on Hopf algebras and quantum
+ groups, and certainly as supplementary reading material for such a
+ course. There are frequent exercises which would be useful for such
+ purposes. Besides being a basic source book, the authors include some
+ new results and some novel approaches to earlier results. All this
+ makes this book a most welcome addition to the quantum group
+ literature."
+}
\end{chunk}
@@ 15403,6 +16142,61 @@ In Anonymous [Ano91], pp287299 (vol. 1) 2 vols.
\end{chunk}
+\index{Letichevskij, A. Alexander}
+\index{Marinchenko, V. G.}
+\begin{chunk}{axiom.bib}
+@article{Leti97,
+ author = "Letichevskij, A. Alexander and Marinchenko, V. G.",
+ title = "Objects in algebraic programming system",
+ journal = "Cybern. Syst. Anal.",
+ volume = "33",
+ number = "2",
+ pages = "283299",
+ year = "1997",
+ keywords = "axiomref",
+ comment = "translated from Russian",
+ abstract =
+ "The algebraic programming system (APS) developed at the
+ V. M. Glushkov Institute of Cybernetics of the Academy of Sciences of
+ the Ukrainian SSR integrates the basic programming paradigms,
+ including procedural, functional, algebraic, and logic programming.
+
+ Algebraic programming in APS relies on special data structures, the
+ socalled graph terms, which permit using diverse data and knowledge
+ representations in relevant application domains. In the language
+ APLAN, graph terms are described by expressions or systems of
+ expressions of a manysorted algebra of data. They may represent both
+ objects of the application domain and reasoning about these
+ objects. The option of setting an arbitrary interpretation of the
+ operations in the algebra of data makes it possible to use APS as a
+ basis for various extensions.
+
+ Symbolic computation systems such as Scratchpad/AXIOM have acquired
+ special importance. They provide various possibilities of manipulating
+ typed mathematical objects, including objects of complex hierarchical
+ structure. This is a natural requirement when working with algebraic
+ objects. In particular, the properties of many algebraic structures
+ (such as groups, rings, fields, etc.) are naturally
+ hierarchicalmodular.
+
+ The Institute of Cybernetics and the Kherson Teachers’ College have
+ developed an instructionoriented computer algebra system AIST. The
+ AIST kernel is a hierarchical structure of mathematical concepts
+ described in the APS language. However, construction of new
+ applications on the basis of this hierarchical structure has proved
+ difficult. The system kernel can be made more flexible by providing
+ tools for flexible description of hierarchical structures of
+ mathematical concepts.
+
+ In this article, we describe an extension of the language APLAN, which
+ provides tools for the objectoriented style of programming. This is
+ one of the possible ways of introducing types in APS. The
+ objectoriented technology also can be used to develop a hierarchical
+ system of mathematical objects."
+}
+
+\end{chunk}
+
\index{Levelt, A. H. M.}
\begin{chunk}{ignore}
\bibitem[Levelt 95]{Lev95} Levelt, A. H. M. (ed)
@@ 16083,13 +16877,19 @@ Invited Presentation in Milestones in Computer Algebra, May 2008, Tobago
\index{Naylor, William A.}
\index{Padget, Julian}
\begin{chunk}{ignore}
\bibitem[Naylor]{NPxx} Naylor, William; Padget, Julian
 title = "From Untyped to Polymorphically Typed Objects in Mathematical Web Services",
+\begin{chunk}{axiom.bib}
+@InProceedings{Nayl06,
+ author = "Naylor, William and Padget, Julian",
+ title = "From Untyped to Polymorphically Typed Objects in Mathematical
+ Web Services",
paper = "NPxx.pdf",
+ series = "Lecture Notes in Computer Science",
+ volume = "4108",
+ pages = "222236",
+ year = "2006",
keywords = "axiomref",
 abstract = "
 OpenMath is a widely recognized approach to the semantic markup of
+ abstract =
+ "OpenMath is a widely recognized approach to the semantic markup of
mathematics that is often used for communication between OpenMath
compliant systems. The Aldor language has a sophisticated
categorybased type system that was specifically developed for the
@@ 16104,6 +16904,7 @@ Invited Presentation in Milestones in Computer Algebra, May 2008, Tobago
service creation and invocation mechanism are explained. Thus we are
in a position to develop and deploy mathematical web services whose
descriptions may be directly derived from Aldor's rich type language."
+}
\end{chunk}
@@ 16613,11 +17414,13 @@ J. of Symbolic Computation 36 pp 513533 (2003)
\end{chunk}
\index{Robidoux, Nicolas}
\begin{chunk}{ignore}
\bibitem[Robidoux 93]{Rob93} Robidoux, Nicolas
+\begin{chunk}{axiom.bib}
+@misc{Robi93,
+ author = "Robidoux, Nicolas",
title = "Does Axiom Solve Systems of O.D.E's Like Mathematica?",
July 1993
 paper = "Rob93.pdf",
+ year = "1993",
+ paper = "Robi93.pdf",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/Robidoux.pdf",
keywords = "axiomref",
abstract = "
If I were demonstrating Axiom and were asked this question, my reply
@@ 16632,6 +17435,7 @@ July 1993
\end{array}
\]
This is a very simple system: $x_1$ is actually uncoupled from $x_2$"
+}
\end{chunk}
@@ 16878,31 +17682,40 @@ in Calmet [Cal94] pp103104
\end{chunk}
\index{Seiler, Werner Markus}
\begin{chunk}{ignore}
\bibitem[Sieler 94b]{Sei94b} Seiler, W.M.
+\begin{chunk}{axiom.bib}
+@article{Seil94b,
+ author = "Seiler, Werner Markus",
title = "Pseudo differential operators and integrable systems in AXIOM",
Computer Physics Communications, 79(2) pp329340 April 1994 CODEN CPHCBZ
ISSN 00104655
 paper = "Sei94b.pdf",
+ journal = "Computer Physics Communications",
+ volume = "79",
+ number = "2",
+ pages = "329340",
+ year = "1994",
+ paper = "Seil94b.pdf",
keywords = "axiomref",
 abstract = "
 An implementation of the algebra of pseudo differential operators in
+ abstract =
+ "An implementation of the algebra of pseudo differential operators in
the computer algebra system Axiom is described. In several exmaples
the application of the package to typical computations in the theory
of integrable systems is demonstrated."
+}
\end{chunk}
\index{Seiler, Werner Markus}
\begin{chunk}{ignore}
\bibitem[Seiler 95]{Sei95} Seiler, W.M.
+\begin{chunk}{axiom.bib}
+@misc{Seil95,
+ author = "Seiler, Werner Markus",
title = "Applying AXIOM to partial differential equations",
Internal Report 9517, Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik
1995
 paper = "Sei95.pdf",
+ institution = {Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik},
+ year = "1995",
+ type = "Internal Report",
+ number = "9517",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/Axiompdf.pdf",
+ paper = "Seil95.pdf",
keywords = "axiomref",
 abstract = "
 We present an Axiom environment called JET for geometric computations
+ abstract =
+ "We present an Axiom environment called JET for geometric computations
with partial differential equations within the framework of the jet
bundle formalism. This comprises expecially the completion of a given
differential equation to an involutive one according to the
@@ 16911,24 +17724,29 @@ Internal Report 9517, Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik
symmetries. Details of the implementations are described and
applications are given. An appendix contains tables of all exported
functions."
+}
\end{chunk}
\index{Seiler, Werner Markus}
\index{Calmet, J.}
\begin{chunk}{ignore}
\bibitem[Seiler 95b]{SC95} Seiler, W.M.; Calmet, J.
 title = "JET  An Axiom Environment for Geometric Computations with Differential Equations",
 paper = "SC95.pdf",
+\begin{chunk}{axiom.bib}
+@misc{Seil95a,
+ author = "Seiler, Werner Markus and Calmet, J.",
+ title = "JET  An Axiom Environment for Geometric Computations with
+ Differential Equations",
+ paper = "Seil95a.pdf",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/axiomjet95.pdf",
keywords = "axiomref",
 abstract = "
 JET is an environment within the computer algebra system Axiom to
+ abstract =
+ "JET is an environment within the computer algebra system Axiom to
perform such computations. The current implementation emphasises the
two key concepts involution and symmetry. It provides some packages
for the completion of a given system of differential equations to an
equivalent involutive one based on the CartanKuranishi theorem and
for setting up the determining equations for classical and
nonclassical point symmetries."
+}
\end{chunk}
@@ 17314,8 +18132,8 @@ IBM Manual, March 1988
keywords = "axiomref",
paper = "Thom01.pdf",
url = "http://axiomwiki.newsynthesis.org/public/refs/aldorcalc2000.pdf",
 abstract = "
 We show how the Aldor type system can represent propositions of
+ abstract =
+ "We show how the Aldor type system can represent propositions of
firstorder logic, by means of the 'propositions as types'
correspondence. The representation relies on type casts (using
pretend) but can be viewed as a prototype implementation of a modified
@@ 17350,12 +18168,16 @@ IBM Manual, March 1988
\end{chunk}
\index{Touratier, Emmanuel}
\begin{chunk}{ignore}
\bibitem[Touratier 98]{Tou98} Touratier, Emmanuel
 title = "Etude du typage dans le syst\`eme de calcul scientifique Aldor",
Universit\'e de Limoges 1998
 paper = "Tou98.pdf",
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Tour98,
+ author = "Touratier, Emmanuel",
+ title = {Etude du typage dans le syst\`eme de calcul scientifique Aldor},
+ comment = "Study of types in the Aldor scientific computation system",
+ year = "1998",
+ paper = "Tour98.pdf",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/AldorT1998_04.pdf",
+ keywords = "axiomref"
+}
\end{chunk}
@@ 17611,13 +18433,43 @@ Version 1.0.0 O($\epsilon{}^1$) June 8, 1994
\index{Broadbery, Peter A.}
\index{Dooley, Sam}
\index{Iglio, Pietro}
\begin{chunk}{ignore}
\bibitem[Watt 94b]{Wat94} Watt, Stephen M.; Broadbery, Peter A.;
Dooley, Samuel S.; Iglio, Pietro
+\begin{chunk}{axiom.bib}
+@techreport{Watt94,
+ author = "Watt, Stephen M. and Broadbery, Peter A. and Dooley, Samuel S.
+ and Iglio, Pietro",
title = "A First Report on the A\# Compiler (including benchmarks)",
IBM Research Report RC19529 (85075) May 12, 1994
 paper = "Wat94.pdf",
+ institution = "IBM Research",
+ year = "1994",
+ type = "technical report",
+ number = "RC19529 (85075)",
+ paper = "Watt94.pdf",
+ url =
+ "http://axiomwiki.newsynthesis.org/public/refs/axiomaldorasharp.pdf",
keywords = "axiomref",
+ abstract =
+ "The $A^{#}$ compiler allows users of computer algebra to develop
+ programs in a context where multiple programming languages are
+ employed. The compiler translates programs written in the $A^{#}$
+ programming language to a lowlevel intermediate language, Foam,
+ from which it can generate standalone programs, native object
+ libraries to be linked with other applications, or code to be read
+ into closed environments. In addition, Foam code may be directly
+ executed using an interpreter provided with the $A^{#}$ compiler.
+
+ The $A^{#}$ programming language provides support for objectoriented
+ and functional programming styles. It is ``higherorder'' in the sense
+ that both types and functions are first class, and may be manipulated
+ in the same ways as any other values. The primary considerations in
+ the formulation of the language have been generality, composibility,
+ and efficiency. The language has been designed to admit a number of
+ important optimizations, allowing compilation to machine code which is
+ in many instances of efficiency comparable to that produced by a C or
+ Fortran compiler.
+
+ The original motivation for $A^{#}$ comes from the field of computer
+ algebra: to provide an improved extension language for the Axiom
+ computer algebra system."
+}
\end{chunk}
@@ 17828,22 +18680,23 @@ ISSAC 94 ACM 0897916387/94/0007
\index{Wester, Michael J.}
\begin{chunk}{axiom.bib}
@misc{West99a,
+@misc{Westxx,
author = "Wester, Michael J.",
 title = "A Critique of the Mathematical Abilities of CA Systems",
 year = "1999",
 url = "http://math.unm.edu/~wester/cas/book/Wester.pdf",
 url2 = "http://math.unm.edu/~wester/cas_review.html",
 paper = "West99a.pdf",
 abstract =
 "Computer algebra systems (CASs) have become an essential computational
 tool in the last decade. General purpose CASs, which are designed to
 solve a wide variety of problems, have gained special prominence. In
 this chapter, the capabilities of seven major general purpose CASs
 (Axiom, Derive, Macsyma, Maple, Mathmatica, MuPAD and Reduce) are
 reviewed on 542 short problems covering a broad range of (primarily)
 symbolic mathematics."

+ title = "Computer Algebra Synonyms",
+ keywords = "axiomref",
+ url = "http://math.unm.edu/~wester/cas/synonyms.pdf",
+ paper = "Westxx.pdf",
+ abstract =
+ "The following is a collection of synonyms for various operations in
+ the seven general purpose computer algebra systems {\bf Axiom}, {\bf
+ Derive}, {\bf Macsyma}, {\bf Maple}, {\bf Mathematica}, {\bf MuPAD},
+ and {\bf Reduce}. This collection does not attempt to be
+ comprehensive, but hopefully it will be useful in giving an indication
+ of how to translate between the syntaxes used by the different systems
+ in many common situations. Note that for a blank entry means that
+ there is no exact translation of a particular operation for the
+ indicated system, but it may still be possible to work around this
+ lack with a related functionality."
}
\end{chunk}
@@ 17855,6 +18708,7 @@ ISSAC 94 ACM 0897916387/94/0007
title = "Computer Algebra Systems. A practical guide",
year = "1999",
publisher = "Wiley",
+ isbn = "0471983535",
keywords = "axiomref",
abstract =
"In this book some of the most popular general purpose computer
@@ 18033,7 +18887,7 @@ D. Reidel Publishing Company H. Werner et. al. (eds.)
\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\index{Zenger, Ch.}
+\index{Zenger, Christoph}
\begin{chunk}{ignore}
\bibitem[Zen92]{Zen92} Zenger, Ch.
title = "Gr{\"o}bnerbasen f{\"u}r Differentialformen und ihre Implementierung in AXIOM'",
@@ 18043,6 +18897,29 @@ Karlsruhe, Germany, 1992
\end{chunk}
+\index{Zenger, Christoph}
+\begin{chunk}{axiom.bib}
+@article{Zeng97,
+ article = "Zenger, Christoph",
+ title = "Indexed types",
+ journal = "Theor. Comput. Sci.",
+ volume = "187",
+ numbers = "12",
+ pages = "147165",
+ year = "1997",
+ keywords = "axiomref",
+ paper = "Zeng97.pdf",
+ abstract =
+ "A new extension of the Hindley/Milner type system is proposed. The
+ type system has algebraic types, that have not only type parameters
+ but also value parameters (indices). This allows for example to
+ parameterize matrices and vectors by their size and to check size
+ compatibility statically. This is especially of interest in computer
+ algebra."
+}
+
+\end{chunk}
+
\index{Zippel, Richard}
\begin{chunk}{ignore}
\bibitem[Zip92]{Zip92} Zippel, Richard
diff git a/changelog b/changelog
index 94a628e..cb9d427 100644
 a/changelog
+++ b/changelog
@@ 1,3 +1,5 @@
+20160629 tpd src/axiomwebsite/patches.html 20160629.01.tpd.patch
+20160629 tpd books/bookvolbib Axiom Citations in the Literature
20160628 tpd src/axiomwebsite/patches.html 20160628.02.tpd.patch
20160628 tpd src/input/Makefile fix typo
20160628 tpd src/axiomwebsite/patches.html 20160628.01.tpd.patch
diff git a/patch b/patch
index 80a06de..285ea86 100644
 a/patch
+++ b/patch
@@ 1,6 +1,1187 @@
src/input/Makefile fix typo
+books/bookvolbib Axiom Citations in the Literature
Goal: Axiom build
+Goal: Axiom Literate Programming
+
+\index{Colin, Antoine}
+\begin{chunk}{axiom.bib}
+@article{Coli97,
+ author = "Colin, Antoine",
+ title = "Solving a system of algebraic equations with symmetries",
+ journal = "J. Pure Appl. Algebra",
+ volume = "117118",
+ pages = "195215",
+ year = "1997",
+ keywords = "axiomref",
+ abstract =
+ "Let $(F)$ be a system of $p$ polynomial equations
+ $F_i({\bf X}) \in k[{\bf X}]$, where $k$ is a commutative field and
+ ${\bf X} := (X_1,\cdots,X_n)$ are indeterminates. Let $G$ be a subgroup
+ of $GL_n(k)$. A polynomial $P \in k[{\bf X}]$ (resp. rational function
+ $P \in k({\bf X})$ ) is an invariant of $G$ if and only if for all
+ $A \in G$ we have $A\cdot P = P$. We denote $k[{\bf X}]^G$ by (resp.
+ $k({\bf X})^G$) the algebra of polynomial (resp. rational function)
+ invariants of $G$. If $L$ is another subgroup of $GL_n(k)$ such that
+ $G \subset L$, $P$ is called a primary invariant of $G$ relative to $L$ if
+ and only if $Stab_L(P) = G$ (where $Stab_L(P)$ is the stabilizer of
+ $P$ in $L$).
+
+ The paper describes the algebra of the invariants of a finite group
+ and how to express these invariants in terms of a small number of
+ them, from both the CohenMacaulay algebra and the field theory points
+ of view. A method is proposed to solve $(F)$ by expressing it in terms of
+ primary invariants $\Pi_1,\cdots,\Pi_n$
+ (e.g. the elementary symmetric polynomials) and one
+ ``primitive'' secondary invariant.
+
+ The main thrust of the paper is contained in the following theorem.
+ Let $(F)$ be a set of invariants of $G$. Let $L$ be a subgroup of
+ $GL_n(k)$ such that $G \subset L$ and $k({\bf X})^L$ is a purely
+ transcendental extension of $k_i$, let $\Pi_1,\cdots,\Pi_n$ be
+ polynomials such that $k({\bf X})^L = k(\Pi_1,\cdots,\Pi_n)$,
+ and let $\Theta \in k[{\bf X}]^G$ be a primitive polynomial invariant
+ of $G$ relative to $L$.
+ When possible, it is convenient to choose $\Theta$ to be one of the
+ polynomials in $(F)$. – An algorithm is given that allows each polynomial
+ $F_i$ to be expressed as $F_i({\bf X}) = H_i(\Pi_1,\cdots,\Pi_n,\Theta)$,
+ an algebraic fraction in $\Pi_1,\cdots,\Pi_n$ and a polynomial in
+ $\Theta$. Now let $L$ be the minimal polynomial of $\Theta$ over
+ $k[{\bf X}]^L$; we have
+ \[L({\bf X},T)=\prod_{\Theta^{'} \in L\cdot \Theta}(T\Theta^{'})
+ \in k[{\bf X}]^L[T]\]
+ (where $L$ is called a generic Lagrange resolvent).
+ As $k(\Pi_1,\cdots,\Pi_n)=k({\bf X})^L$, we can write
+ $L({\bf X},T)=H_0(\Pi_1,\cdots,\Pi_n,T)$ where $H_0$ is some
+ rational function. The question
+ $H_0(\Pi_1,\cdots,\Pi_n,\Theta)=0$ is always satisfied because
+ $\Theta$ is a root of $L$. Then, we solve the system of ($p=1$)
+ algebraic equations $H_i(\Pi_1,\cdots,\Pi_n,\Theta)=0$,
+ $0 \le i \le p$ for $\Pi_1,\cdots,\Pi_n,\Theta$ as indeterminates.
+
+ Theorem 1: Let $D \in k[\Pi_1,\cdots,\Pi_n]$ be the LCM of the
+ denominators of all the fractions $H_i$,$0 \le i \le p$ and let
+ $H_i^{'}=DH_i$. For every solution
+ $x:=(x_1,\cdots,x_n)$ of the system $(F)$:$F_i({\bf X})=0$,
+ $1 \le i \le p$, there exists a solution ($\pi_1,\cdots,\pi_n,\Theta$)
+ of the system
+ $(H^{'}):H_i^{'}(\Pi_1,\cdots,\Pi_n,\Theta)=0$, $0 \le i \le p$ such
+ that $x$ is a solution of the system
+ $(P_\pi):\Pi_i({\bf X})=\pi_i$, $1 \le i \le n$ , and of the equation
+ $\Theta({\bf X})=0$. Conversely, for any solution
+ $(\[i_1,\cdots,\pi_n,\theta)$ of the system $(H^{'})$ such that
+ $D(\pi_1,\cdots,\pi_n) \ne 0$, if $x$ is a solution of the system
+ $(P_\pi)$ relative to $(\pi_1,\cdots,\pi_n)$, then there exists
+ some $A \in L$ such that $\Theta(A\cdot x)=\theta$, and then for all
+ $B \in G$, $BA\cdot x$, is a solution of the system $(F)$.
+
+ A slighly more general version of this theorem is also given. The
+ paper then presents an algorithm that applies the theory and has been
+ implemented in AXIOM. It is followed by several examples."
+}
+
+\end{chunk}
+
+\index{DiBlasio, Paolo}
+\index{Temperini, Marco}
+\begin{chunk}{axiom.bib}
+@article{DiBl95,
+ author = "DiBlasio, Paolo and Temperini, Marco",
+ title = "Subtyping Inheritance and Its Application in Languages for
+ Symbolic Computation Systems",
+ journal = "J. Symbolic Computation",
+ volume = "19",
+ pages = "3963",
+ year = "1995",
+ paper = "DiBl95.pdf",
+ keywords = "axiomref",
+ abstract =
+ "Application of objectoriented programming techniques to design and
+ implementation of symbolic computation is investigated. We show the
+ significance of certain correctness problems, occurring in programming
+ environments based on specialization inheritance, due to use of method
+ redefinition and polymorphism. We propose a solution to these
+ problems, by defining a mechanism of subtyping inheritance and the
+ prototype of an objectoriented programming language for a symbolic
+ computation system. We devise the subtyping inheritance {\sl ESI
+ (Enhanced String Inheritance)} by lifting to programming language
+ constructs a given model of subtyping, which is established by a
+ monotonic (covariant) subtyping rule. Type safeness of language
+ instructions is proved.
+
+ The adoption of {\sl ESI} allows to model method and class
+ specialization in a natural way. The {\sl ESI} mechanism verifies the
+ type correctness of language statements by means of type checking
+ rules and preserves their correctness at runtime by a suitable method
+ lookup algorithm."
+}
+
+\end{chunk}
+
+\index{DiBlasio, Paolo}
+\index{Temperini, Marco}
+\begin{chunk}{axiom.bib}
+@InProceedings{DiBl97,
+ author = "DiBlasio, Paolo and Temperini, Marco",
+ title = "On subtyping in languages for symbolic computation systems",
+ booktitle = "Advances in the design of symbolic computation systems",
+ series = "Monographs in Symbolic Computation",
+ year = "1997",
+ publisher = "Springer",
+ pages = "164178",
+ keywords = "axiomref",
+ abstract =
+ "We want to define a strongly typed OOP language suitable as the
+ software development tool of a symbolic computation system, which
+ provides class structure to manage ADTs and supports multiple
+ inheritance to model specialization hierarchies. In this paper, we
+ provide the theoretical background for such a task."
+}
+
+\end{chunk}
+
+\index{Fakler, Winfried}
+\begin{chunk}{axiom.bib}
+@article{Fakl97,
+ author = "Fakler, Winfried",
+ title = "On second order homogeneous linear differential equations with
+ Liouvillian solutions",
+ journal = "Theor. Comput. Sci.",
+ volume = "187",
+ number = "12",
+ pages = "2748",
+ year = "1997",
+ paper = "Fakl97.pdf",
+ keywords = "axiomref",
+ abstract =
+ "We determine all minimal polynomials for second order homogeneous
+ linear differential equations with algebraic solutions decomposed into
+ invariants and we show how easily one can recover the known conditions
+ on differential Galois groups [J. Kovacic, J. Symb. Comput. 2, 343
+ (1986; Zbl 0603.68035), M. F. Singer and F. Ulmer,
+ J. Symb. Comput. 16, 936, 3773 (1993; Zbl 0802.12004, Zbl
+ 0802.12005), F.Ulmer and J. A. Weil, J. Symb. Comput. 22, 179200
+ (1996; Zbl 0871.12008)] using invariant theory. Applying these
+ conditions and the differential invariants of a differential equation
+ we deduce an alternative method to the algorithms given in (loc. cit.)
+ for computing Liouvillian solutions. For irreducible second order
+ equations our method determines solutions by formulas in all but three
+ cases."
+}
+
+\end{chunk}
+
+\index{Jacquemard, Alain}
+\index{KhechichineMourtada, F.Z.}
+\index{Mourtada, A.}
+\begin{chunk}{axiom.bib}
+@article{Jacq97,
+ author = "Jacquemard, Alain and KhechichineMourtada, F.Z. and Mourtada, A.",
+ title = "Formal algorithms applied to the study of the cyclicity of a
+ generic algebraic polycycle with four hyperbolic crests",
+ journal = "Nonlinearity",
+ volume = "10",
+ number = "1",
+ pages = "1953",
+ year = "1997",
+ keywords = "axiomref",
+ comment = "french",
+ abstract =
+ "Drawing on the work of Mourtada, we show that a family of vector
+ fields with a generic algebraic polycycle of four hyperbolic apices
+ possesses a maximum capacity of four limit cycles. This cyclicity is
+ attained in an opening connecting the parameters which the edge
+ contains, in particular a generic line of singularities of dovetail
+ type. We also give an asymptotic estimation of the volume of this
+ opening, as well as an explicit example of a family of polynomial
+ vector fields replicating the abovedescribed conditions and
+ possessing five limit cycles. The methods employed are very diverse:
+ geometrical arguments (Thom’s theory of catastrophes and the theory of
+ algebraic singularities), developments from Puiseux, the number of
+ major roots by Descartes’ law and calculated exactly by Sturm series,
+ and other specific methods for formal calculus, such as for example
+ the cylindrical algebraic decomposition and the resolution of
+ algebraic systems via the construction of Gröbner bases. The
+ calculations have been executed formally, that is to say without
+ making the least appeal to numerical approximation, in using the
+ formal calculus system AXIOM."
+}
+
+\end{chunk}
+
+\index{Lambe, Larry A.}
+\index{Radford, David E.}
+\begin{chunk}{axiom.bib}
+@book{Lamb97,
+ author = "Lambe, Larry A. and Radford, David E.",
+ title = "Introduction to the quantum YangBaxter equation and quantum
+ groups: an algebraic approach",
+ booktitle = "Mathematics and its Applications",
+ publisher = "Kluwer Adademic Publishers",
+ year = "1997",
+ keywords = "axiomref",
+ abstract =
+ "The quantum YangBaxter equation (QYBE) has roots in statistical
+ mechanics and the inverse scattering method and leads to a natural
+ construction of a bialgebra. It turns out to have important
+ connections with knot theory and invariants of 3manifolds. There are
+ now available many reference books to quantum groups and these various
+ applications. The book under review develops the algebraic
+ underpinning and theory of the QYBE, including the constant form and
+ the one and two parameter forms.
+
+ We give a brief description of the chapters. Chapter 1 (together with
+ an Appendix) gives the algebraic preliminaries involving coalgebras,
+ bialgebras, Hopf algebras, modules and comodules. Chapter 2 introduces
+ the various forms of the QYBE, and the basic algebraic structures
+ associated to them, including FaddeevReshetikhinTakhtadzhan (FRT)
+ construction. Chapter 3 explores various categorical settings for the
+ constant form of the QYBE, the most basic being the category of left
+ QYB modules over a bialgebra and the notion of algebras, coalgebras,
+ etc. in this category. Chapter 4 develops universal mapping properties
+ of the FRT construction and its reduced version, and the authors
+ investigate when the reduced FRT construction leads to a pointed
+ bialgebra or a pointed Hopf algebra. Chapter 5 develops the quantum
+ groups associated to $SL(2)$, i.e., the quantum universal enveloping
+ algebra, and the quantum function algebra. Chapter 6 introduces
+ quasitriangular Hopf algebras, and discusses how the
+ finitedimensional ones give rise to solutions of the QYBE through
+ their representation theory. The most important example is the
+ Drinfeld double of a finitedimensional Hopf algebra. The authors note
+ (through an exercise!) that every finitedimensional Hopf algebra is
+ the reduced FRT construction of some solution to the QYBE. Chapter 7
+ introduces coquasitriangular bialgebras, the most important being the
+ FRT and the reduced FRT constructions. There are some generalizations
+ here to the oneparameter form of the QYBE. Chapter 8 uses all the
+ previously developed techniques to find solutions of the QYBE in
+ certain cases, including the oneparameter form. Some of these were
+ discovered by computer algebra methods. The final chapter 9 gives a
+ brief discussion of certain categorical constructions and the QYBE is
+ certain fairly abstract categories, motivated by the fact that the FRT
+ construction is a coend.
+
+ This book fills an important niche in the literature involving the
+ QYBE by highlighting the algebraic aspects and applications. Although
+ this is basically a reference book, it includes so many important
+ parts of the study of Hopf algebras that it could be used as a
+ textbook for a certain type of course on Hopf algebras and quantum
+ groups, and certainly as supplementary reading material for such a
+ course. There are frequent exercises which would be useful for such
+ purposes. Besides being a basic source book, the authors include some
+ new results and some novel approaches to earlier results. All this
+ makes this book a most welcome addition to the quantum group
+ literature."
+}
+
+\end{chunk}
+
+\index{Letichevskij, A. Alexander}
+\index{Marinchenko, V. G.}
+\begin{chunk}{axiom.bib}
+@article{Leti97,
+ author = "Letichevskij, A. Alexander and Marinchenko, V. G.",
+ title = "Objects in algebraic programming system",
+ journal = "Cybern. Syst. Anal.",
+ volume = "33",
+ number = "2",
+ pages = "283299",
+ year = "1997",
+ keywords = "axiomref",
+ comment = "translated from Russian",
+ abstract =
+ "The algebraic programming system (APS) developed at the
+ V. M. Glushkov Institute of Cybernetics of the Academy of Sciences of
+ the Ukrainian SSR integrates the basic programming paradigms,
+ including procedural, functional, algebraic, and logic programming.
+
+ Algebraic programming in APS relies on special data structures, the
+ socalled graph terms, which permit using diverse data and knowledge
+ representations in relevant application domains. In the language
+ APLAN, graph terms are described by expressions or systems of
+ expressions of a manysorted algebra of data. They may represent both
+ objects of the application domain and reasoning about these
+ objects. The option of setting an arbitrary interpretation of the
+ operations in the algebra of data makes it possible to use APS as a
+ basis for various extensions.
+
+ Symbolic computation systems such as Scratchpad/AXIOM have acquired
+ special importance. They provide various possibilities of manipulating
+ typed mathematical objects, including objects of complex hierarchical
+ structure. This is a natural requirement when working with algebraic
+ objects. In particular, the properties of many algebraic structures
+ (such as groups, rings, fields, etc.) are naturally
+ hierarchicalmodular.
+
+ The Institute of Cybernetics and the Kherson Teachers’ College have
+ developed an instructionoriented computer algebra system AIST. The
+ AIST kernel is a hierarchical structure of mathematical concepts
+ described in the APS language. However, construction of new
+ applications on the basis of this hierarchical structure has proved
+ difficult. The system kernel can be made more flexible by providing
+ tools for flexible description of hierarchical structures of
+ mathematical concepts.
+
+ In this article, we describe an extension of the language APLAN, which
+ provides tools for the objectoriented style of programming. This is
+ one of the possible ways of introducing types in APS. The
+ objectoriented technology also can be used to develop a hierarchical
+ system of mathematical objects."
+}
+
+\end{chunk}
+
+\index{Schwarzweller, Christoph}
+\begin{chunk}{axiom.bib}
+@phdthesis{Schw97,
+ author = "Schwarzweller, Christoph",
+ title = "MIZAR verification of generic algebraic algorithms",
+ school = "University of Tubingen",
+ year = "1997",
+ paper = "Schw97.pdf",
+ keywords = "axiomref",
+ abstract =
+ "Although generic programming founds more and more attention –
+ nowadays generic programming languages as well as generic libraries
+ exist – there are hardly approaches for the verification of generic
+ algorithms or generic libraries. This thesis deals with generic
+ algorithms in the field of computer algebra. We propose the Mizar
+ system as a theorem prover capable of verifying generic algorithms on
+ an appropriate abstract level. The main advantage of the MIZAR theorem
+ prover is its special input language that enables textbook style
+ presentation of proofs. For generic versions of Brown/Henrici addition
+ and of Euclidean’s algorithm we give complete correctness proofs
+ written in the MIZAR language.
+
+ Moreover, we do not only prove algorithms correct in the usual
+ sense. In addition we show how to check, using the MIZAR system, that
+ a generic algebraic algorithm is correctly instantiated with a
+ particular domain. Answering this question that especially arises if
+ one wants to implement generic programming languages, in the field of
+ computer algebra requires nontrivial mathematical knowledge.
+
+ To build a verification system using the MIZAR theorem prover, we also
+ implemented a generator which almost automatically computes for a
+ given algorithm a set of theorems that imply the correctness of this
+ algorithm."
+}
+
+\end{chunk}
+
+\index{Zenger, Christoph}
+\begin{chunk}{axiom.bib}
+@article{Zeng97,
+ article = "Zenger, Christoph",
+ title = "Indexed types",
+ journal = "Theor. Comput. Sci.",
+ volume = "187",
+ numbers = "12",
+ pages = "147165",
+ year = "1997",
+ keywords = "axiomref",
+ paper = "Zeng97.pdf",
+ abstract =
+ "A new extension of the Hindley/Milner type system is proposed. The
+ type system has algebraic types, that have not only type parameters
+ but also value parameters (indices). This allows for example to
+ parameterize matrices and vectors by their size and to check size
+ compatibility statically. This is especially of interest in computer
+ algebra."
+}
+
+\end{chunk}
+
+\index{Bernardin, Laurent}
+\begin{chunk}{axiom.bib}
+@article{Bern96,
+ author = "Benardin, Laurent",
+ title = "A review of symbolic solvers",
+ journal = "SIGSAM Bull.",
+ volume = "30",
+ number = "1",
+ pages = "920",
+ year = "1996",
+ keywords = "axiomref",
+ paper = "Bern96.pdf",
+ abstract =
+ "Solving equations and systems of equations symbolically is a key
+ feature of every Computer Algebra System. This review examines the
+ capabilities of the six best known general purpose systems to date in
+ the area of general algebraic and transcendental equation
+ solving. Areas explicitly not covered by this review are differential
+ equations and numeric or polynomial system solving as special purpose
+ systems exist for these kinds of problems. The aim is to provide a
+ benchmark for comparing Computer Algebra Systems in a specific
+ domain. We do not intend to give a rating of overall capabilities as
+ for example in [9]. 1 The Contestants We compare six major Computer
+ Algebra Systems. Axiom 2.0 [7], Derive 3.06 [1], Macsyma 420 [8],
+ Maple V R4 [3], Mathematica 2.2 [10], MuPAD 1.2.9 [5] and Reduce 3.6
+ [6]. When available, we tried to use the latest shipping version of
+ each system. 2 The Problem Set The following table presents the set of
+ 80 problems that we used to evaluate the different solvers..."
+}
+
+\end{chunk}
+
+\index{Wester, Michael J.}
+\begin{chunk}{axiom.bib}
+@misc{Westxx,
+ author = "Wester, Michael J.",
+ title = "Computer Algebra Synonyms",
+ keywords = "axiomref",
+ url = "http://math.unm.edu/~wester/cas/synonyms.pdf",
+ paper = "Westxx.pdf",
+ abstract =
+ "The following is a collection of synonyms for various operations in
+ the seven general purpose computer algebra systems {\bf Axiom}, {\bf
+ Derive}, {\bf Macsyma}, {\bf Maple}, {\bf Mathematica}, {\bf MuPAD},
+ and {\bf Reduce}. This collection does not attempt to be
+ comprehensive, but hopefully it will be useful in giving an indication
+ of how to translate between the syntaxes used by the different systems
+ in many common situations. Note that for a blank entry means that
+ there is no exact translation of a particular operation for the
+ indicated system, but it may still be possible to work around this
+ lack with a related functionality."
+}
+
+\end{chunk}
+
+\index{Wester, Michael J.}
+\begin{chunk}{axiom.bib}
+@misc{West95,
+ author = "Wester, Michael J.",
+ title = "A Review of CAS Mathematical Capabilities",
+ year = "1995",
+ keywords = "axiomref",
+ paper = "West95.pdf",
+ url = "http://math.unm.edu/~wester/cas/Paper.ps",
+ abstract =
+ "Computer algebra systems (CASs) have become an important
+ computational tool in the last decade. General purpose CASs, which are
+ designed to solve a wide variety of problems, have gained special
+ prominance. In this paper, the capabilities of seven major general
+ purpose CASs (Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, and
+ Reduce) are reviewed on 131 short problems covering a broad range of
+ (primarily) symbolic mathematics.
+
+ A demo was developed for each CAS, run and the results
+ evaluated. Problems were graded in terms of whether it was easy or
+ difficult or possible to produce an answer and if an answer was
+ produced, whether it was correct. It is the author's hope that this
+ review will encourage the development of a comprehensive CAS test
+ suite."
+}
+
+\end{chunk}
+
+\index{Apel, Joachim}
+\index{Klaus, Uwe}
+\begin{chunk}{axiom.bib}
+@misc{Apel94,
+ author = "Apel, Joachim and Klaus, Uwe",
+ title = "Representing Polynomials in Computer Algebra Systems",
+ year = "1994",
+ paper = "Apel94.pdf",
+ abstract =
+ "There are discussed implementational aspects of the specialpurpose
+ computer algebra system FELIX designed for computations in
+ constructive algebra. In particular, data types developed for the
+ representation of and computation with commutative and noncommuative
+ polynomials are described. Furthermore, comparison of time and memory
+ requirements of different polynomial representations are reported."
+}
+
+\end{chunk}
+
+\index{Stoutemyer, David R.}
+\begin{chunk}{axiom.bib}
+@article{Stou91,
+ author = "Stoutemyer, David R.",
+ title = "Crimes and misdemeanors in the computer algebra trade",
+ journal = "Notices of the American Mathematical Society",
+ volume = "38",
+ number = "7",
+ pages = "778785",
+ year = "1991"
+}
+
+\end{chunk}
+
+\index{Sangwin, Chris}
+\begin{chunk}{axiom.bib}
+@misc{Sang10,
+ author = "Sangwin, Chris",
+ title = "Intriguing Integrals: Part I and II",
+ year = "2010",
+ url1 =
+ "https://plus.maths.org/issue54/features/sangwin/2pdf/index.html/op.pdf",
+ paper1 = "Sang10a.pdf",
+ url2 =
+ "https://plus.maths.org/issue54/features/sangwin2/2pdf/index.html/op.pdf",
+ paper2 = "Sang10b.pdf"
+}
+
+\end{chunk}
+
+\index{Evans, Brian}
+\begin{chunk}{axiom.bib}
+@misc{Evanxx,
+ author = "Evans, Brian",
+ title = "History of CA Systems",
+ url = "http://felix.unife.it/Root/dMathematics/dThemathematician/dHistoryofmathematics/tHistoryofcomputeralgebra",
+ paper = "Evanxx.txt"
+}
+
+\end{chunk}
+
+\index{Martin, Ursula}
+\index{Shand, D.}
+\begin{chunk}{axiom.bib}
+@misc{Mart97,
+ author = "Martin, Ursula and Shand, D",
+ title = "Investigating some Embedded Verification Techniques for
+ Computer Algebra Systems",
+ url = "http://www.risc.jku.at/conferences/Theorema/papers/shand.ps.gz",
+ paper = "Mart97.ps",
+ abstract = "
+ This paper reports some preliminary ideas on a collaborative project
+ between St. Andrews University in the UK and NAG Ltd. The project aims
+ to use embedded verification techniques to improve the reliability and
+ mathematical soundness of computer algebra systems. We give some
+ history of attempts to integrate computer algebra systems and
+ automated theorem provers and discuss possible advantages and
+ disadvantages of these approaches. We also discuss some possible case
+ studies."
+}
+
+\end{chunk}
+
+\index{Tonisson, Eno}
+\begin{chunk}{axiom.bib}
+@article{Tonixx,
+ author = "Tonisson, Eno",
+ title = "Branch Completeness in School Mathematics and in Computer Algebra
+ Systems",
+ journal = "The Electronic Journal of Mathematics and Technology",
+ volume = "1",
+ number = "1",
+ issn = "19332823",
+ paper = "Tonixx.pdf",
+ url = "https://php.radford.edu/~ejmt/deliveryBoy.php?paper=eJMT_v1n3p5",
+ abstract =
+ "In many cases when solving school algebra problems (e.g. simplifying
+ an expression, solving an equation), the solution is separable into
+ branches in some manner. The paper describes some approaches to
+ branches that are used in school textbooks and computer algebra
+ systems and compares them with mathematically branchcomplete
+ solutions. It tries to identify possible reasons behind different
+ approaches and also indicate some ideas how such differences could be
+ explained to the students."
+}
+
+\end{chunk}
+
+\index{Beeson, Michael}
+\begin{chunk}{axiom.bib}
+@misc{Beesxx,
+ author = "Beeson, Michael",
+ title = "Automatic Generation of EpsilonDelta Proofs of Continuity",
+ url = "http://www.michaelbeeson.com/research/papers/aisc.pdf",
+ paper = "Beesxx.pdf",
+ abstract =
+ "As part of a project on automatic generation of proofs involving both
+ logic and computation, we have automated the production of some proofs
+ involving epsilondelta arguments. These proofs involve two or three
+ quantifiers on the logical side, and on the computational side, they
+ involve algebra, trigonometry, and some calculus. At the border of
+ logic and computation, they involve several types of arguments
+ involving inequalities, including transitivity chaining and several
+ types of bounding arguments, in which bounds are sought that do not
+ depend on certain variables. Control mechanisms have been developed
+ for intermixing logical deduction steps with computational steps and
+ with inequality reasoning. Problems discussed here as examples involve
+ the continuity and uniform continuity of various specific functions."
+}
+
+\end{chunk}
+
+\index{Ballarin, Clemens}
+\index{Paulson, Lawrence C.}
+\begin{chunk}
+@misc{Ball98,
+ author = "Ballarin, Clemens and Paulson, Lawrence C.",
+ title = "Reasoning about Coding Theory: The Benefits We Get from
+ Computer Algebra",
+ year = "1998",
+ url = http://www21.in.tum.de/~ballarin/publications/aisc98.pdf",
+ paper = "Ball98.pdf",
+ abstract =
+ "The use of computer algebra is usually considered beneficial for
+ mechanised reasoning in mathematical domains. We present a case study,
+ in the application domain of coding theory, that supports this claim:
+ the mechanised proof depends on nontrivial algorithms from computer
+ algebra and increase the reasoning power of the theorem prover. The
+ unsoundness of computer algebra systems is a major problem in
+ interfacing them to theorem provers. Our approach to obtaining a sound
+ overall system is not blanket distrust but based on the distinction
+ between algorithms we call sound and {\sl ad hoc} respectively. This
+ distinction is blurred in most computer algebra systems OUr
+ experimental interface therefore uses a computer algebra library. It
+ is based on theorem templates, which provide formal specifications for
+ the algorithms."
+}
+
+\end{chunk}
+
+\index{Aslaksen, Helmer}
+\begin{chunk}{axiom.bib}
+@article{Asla96,
+ author = "Aslaksen, Helmer",
+ title = "Multiplevalued complex functions and computer algebra",
+ journal = "SIGSAM Bulletin",
+ volume = "30",
+ number = "2",
+ year = "1996",
+ pages = "1220",
+ paper = "Asla96.pdf",
+ url = "http://www.math.nus.edu.sg/aslaksen/papers/cacas.pdf",
+ abstract =
+ "I recently taught a course on complex analysis. That forced me to
+ think more carefully about branches. Being interested in computer
+ algebra, it was only natural that I wanted to see how such programs
+ dealt with these problems. I was also inspired by a paper by
+ Stoutemyer.
+
+ While programs like Derive, Maple, Mathematica and Reduce are very
+ powerful, they also have their fair share of problems. In particular,
+ branches are somewhat of an Achilles' heel for them. As is wellknown,
+ the complex logarithm function is properly defined as a
+ multiplevalued function. And since the general power and exponential
+ functions are defined in terms of the logarithm function, they are
+ also multiplevalued. But for actual computations, we need to make
+ them single valued, which we do by choosing a branch. In Section 2, we
+ will consider some transformation rules for branches of
+ multiplevalued complex functions in painstaking detail.
+
+ The purpose of this short article is not to do a comprehensive
+ comparative study of different computer algebra systems. My goal is
+ simply to make the readers aware of some of the problems, and to
+ encourage the readers to sit down and experiment with their favourite
+ programs."
+}
+
+\end{chunk}
+
+\index{Fateman, Richard J.}
+\begin{chunk}{axiom.bib}
+@InProceedings{Fate96,
+ author = "Fateman, Richard J.",
+ title = "A Review of Symbolic Solvers",
+ booktitle = "Proc 1996 ISSAC",
+ series = "ISSAC 96",
+ year = "1996",
+ pages = "8694",
+ keywords = "axiomref",
+ keywords = "axiomref",
+ paper = "Fate96.pdf",
+ url = "http://http.cs.berkeley.edu/~fateman/papers/eval.ps",
+ abstract =
+ "``Evaluation'' of expressions and programs in a computer algebra
+ system is central to every system, but inevitably fails to provide
+ complete satisfaction. Here we explain the conflicting requirements,
+ describe some solutions from current systems, and propose alternatives
+ that might be preferable sometimes. We give examples primarily from
+ Axiom, Macsyma, Maple, Mathematica, with passing metion of a few other
+ systems."
+}
+
+\end{chunk}
+
+\index{Fateman, Richard J.}
+\begin{chunk}{axiom.bib}
+@misc{Fate05,
+ author = "Fateman, Richard J.",
+ title = "An incremental approach to building a mathematical
+ expert out of software",
+ conference = "Axiom Computer Algebra Conference",
+ location = "City College of New York, CAISS project",
+ year = "2005",
+ month = "April",
+ day = "19",
+ url = "http://www.cs.berkeley.edu/~fateman/papers/axiom.pdf",
+ paper = "Fat05.pdf",
+ keywords = "axiomref"
+}
+
+\end{chunk}
+
+\index{Gr\"abe, HansGert}
+\begin{chunk}{axiom.bib}
+@misc{Grab98,
+ author = "Grabe, HansGert",
+ title = "About the Polynomial System Solve Facility of Axiom, Macsyma,
+ Maple Mathematica, MuPAD, and Reduce",
+ paper = "Grab98.pdf",
+ url =
+"https://www.informatik.unileipzig.de/~graebe/ComputerAlgebra/Publications/WesterBook.pdf",
+ keywords = "axiomref",
+ abstract =
+ "We report on some experiences with the general purpose Computer
+ Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and
+ Reduce solving systems of polynomial equations and the way they
+ present their solutions. This snapshot (taken in the spring of 1996)
+ of the current power of the different systems in a special area
+ concentrates on both CPUtimes and the quality of the output."
+}
+
+\end{chunk}
+
+\index{Gr\"abe, HansGert}
+\begin{chunk}{axiom.bib}
+@misc{Grab06,
+ author = "Grabe, HansGert",
+ title = "The Groebner Factorizer and Polynomial System Solving",
+ year = "2006",
+ keywords = "axiomref",
+ report = "Special Semester on Groebner Bases",
+ location = "Linz",
+ paper = "Grab06.pdf",
+ url =
+"https://www.ricam.oeaw.ac.at/specsem/srs/groeb/download/06\_02\_Solver.pdf",
+ abstract =
+ "Let $S := k[x_1,\ldots, x_n]$ be the polynomial ring in the
+ variables $x_1,\ldots,x_n$ over the field $k$ and
+ $B := \{f_1,\ldots,f_m\} \subset S$
+ be a finite system of polynomials. Denote by $I(B)$ the
+ ideal generated by these polynomials. One of the major tasks of
+ constructive commutative algebra is the derivation of information
+ about the structure of
+ \[V(B):=\{a \in K^n : \forall f \in B{\rm\ such\ that\ }f(a)=0\}\]
+ the set of common zeroes of the system $B$ over an
+ algebraically closed extension $K$ of $k$. Splitting the system into
+ smaller ones, solving them separately, and patching all solutions
+ together is often a good guess for a quick solution of even highly
+ nontrivial problems. This can be done by several techniques, e.g.,
+ characteristic sets, resultants, the Groebner factorizer or some ad
+ hoc methods. Of course, such a strategy makes sense only for problems
+ that really will split, i.e., for reducible varieties of
+ solutions. Surprisingly often, problems coming from 11real life''
+ fulfill this condition.
+
+ Among the methods to split polynomial systems into smaller pieces
+ probably the Groebner factor izer method attracted the most
+ theoretical attention, see Czapor ([4, 5]), Davenport ([6]), Melenk, M
+ ̈oller and Neun ([16, 17]) and Gr ̈abe ([13, 14]). General purpose
+ Computer Algebra Systems (CAS) are well suited for such an approach,
+ since they make available both a (more or less) well tuned
+ implementation of the classical Groebner algorithm and an effective
+ multivariate polynomial factorizer.
+
+ Furthermore it turned out that the Groebner factorizer is not only a
+ good heuristic approach for splitting, but its output is also usually
+ a collection of almost prime components. Their description allows a
+ much deeper understanding of the structure of the set of zeroes
+ compared to the result of a sole Groebner basis computation.
+
+ Of course, for special purposes a general CAS as a multipurpose
+ mathematical assistant can’t offer the same power as specialized
+ software with efficiently implemented and well adapted algorithms and
+ data types. For polynomial system solving, such specialized software
+ has to implement two algorithmically complex tasks, solving and
+ splitting, and until recently none of the specialized systems (as
+ e.g., GB, Macaulay, Singular, CoCoA, etc.) did both
+ efficiently. Meanwhile, being very efficient computing (classical)
+ Groebner bases, development efforts are also directed, not only
+ for performance reasons, towards a better inclusion of factorization
+ into such specialized systems. Needless to remark that it needs some
+ skill to force a special system to answer questions and the user will
+ probably first try his ``home system'' for an answer. Thus the
+ polynomial systems solving facility of the different CAS should behave
+ especially well on such polynomial systems that are hard enough not to
+ be done by hand, but not really hard to require special efforts. It
+ should invoke a convenient interface to get the solutions in a form
+ that is (correct and) well suited for further analysis in the familiar
+ environment of the given CAS as the personal mathematical assistant."
+}
+
+\end{chunk}
+
+\index{Corless, Robert M.}
+\index{Jeffrey, David J.}
+\index{Watt, Stephen M.}
+\index{Bradford, Russell}
+\index{Davenport, James H.}
+\begin{chunk}{axiom.bib}
+@misc{Corl0,
+ author = "Corless, Robert M. and Jeffrey, David J. and Watt, Stephen M.
+ and Bradford, Russell and Davenport, James H.",
+ title = "Reasoning about the elementary functions of complex analysis",
+ url = "http://www.csd.uwo.ca/~watt/pub/reprints/2002amaireasoning.pdf",
+ paper = "Corl05.pdf",
+ abstract = "
+ There are many problems with the simplification of elementary
+ functions, particularly over the complex plane. Systems tend to make
+ ``howlers'' or not to simplify enough. In this paper we outline the
+ ``unwinding number'' approach to such problems, and show how it can be
+ used to prevent errors and to systematise such simplification, even
+ though we have not yet reduced the simplification process to a
+ complete algorithm. The unsolved problems are probably more amenable
+ to the techniques of artificial intelligence and theorem proving than
+ the original problem of complexvariable analysis."
+}
+
+\end{chunk}
+
+\index{Touratier, Emmanuel}
+\begin{chunk}{axiom.bib}
+@misc{Tour98,
+ author = "Touratier, Emmanuel",
+ title = {Etude du typage dans le syst\`eme de calcul scientifique Aldor},
+ comment = "Study of types in the Aldor scientific computation system",
+ year = "1998",
+ paper = "Tour98.pdf",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/AldorT1998_04.pdf",
+ keywords = "axiomref"
+}
+
+\end{chunk}
+
+\index{Seiler, Werner Markus}
+\begin{chunk}{axiom.bib}
+@misc{Seil95,
+ author = "Seiler, Werner Markus",
+ title = "Applying AXIOM to partial differential equations",
+ institution = {Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik},
+ year = "1995",
+ type = "Internal Report",
+ number = "9517",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/Axiompdf.pdf",
+ paper = "Seil95.pdf",
+ keywords = "axiomref",
+ abstract =
+ "We present an Axiom environment called JET for geometric computations
+ with partial differential equations within the framework of the jet
+ bundle formalism. This comprises expecially the completion of a given
+ differential equation to an involutive one according to the
+ CartanKuranishi Theorem and the setting up of the determining system
+ for the generators of classical and nonclassical Lie
+ symmetries. Details of the implementations are described and
+ applications are given. An appendix contains tables of all exported
+ functions."
+}
+
+\end{chunk}
+
+\index{Davenport, James H.}
+\begin{chunk}{axiom.bib}
+@misc{Dave84a,
+ author = "Davenport, James H.",
+ title = "A New Algebra System",
+ paper = "Dave84a.pdf",
+ keywords = "axiomref",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/Davenport1984a\_new\_algebra\_system.pdf",
+ abstract =
+ "Seminal internal paper discussing Axiom design decisions."
+}
+
+\end{chunk}
+
+\index{Conrad, Marc}
+\index{French, Tim}
+\index{Maple, Carsten}
+\index{Pott, Sandra}
+\begin{chunk}{axiom.bib}
+@misc{Conrxxa,
+ author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra",
+ title = "Approaching Inheritance from a Natural Mathematical Perspective
+ and from a Java Driven Viewpoint: a Comparative Review",
+ keywords = "axiomref",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/McTfCmSpaxiom.pdf",
+ paper = "Conrxxa.pdf",
+ abstract = "
+ It is wellknown that few objectoriented programming languages allow
+ objects to change their nature at runtime. There have been a number
+ of reasons presented for this, but it appears that there is a real
+ need for matters to change. In this paper we discuss the need for
+ objectoriented programming languages to reflect the dynamic nature of
+ problems, particularly those arising in a mathematical context. It is
+ from this context that we present a framework that realistically
+ represents the dynamic and evolving characteristic of problems and
+ algorithms."
+}
+
+\end{chunk}
+
+\index{Meijer, Erik}
+\index{Fokkinga, Maarten}
+\index{Paterson, Ross}
+\begin{chunk}{axiom.bib}
+@misc{Meij91,
+ author = "Meijer, Erik and Fokkinga, Maarten and Paterson, Ross",
+ title = "Functional Programming with Bananas, Lenses, Envelopes and
+ Barbed Wire",
+ url = "http://eprints.eemcs.utwente.nl/7281/01/dbutwente40501F46.pdf",
+ paper = "Meij91.pdf",
+ abstract = "
+ We develop a calculus for lazy functional programming based on
+ recursion operators associated with data type definitions. For these
+ operators we derive various algebraic laws that are useful in deriving
+ and manipulating programs. We shall show that all example functions in
+ Bird and Wadler's ``Introduction to Functional Programming'' can be
+ expressed using these operators."
+}
+
+\end{chunk}
+
+\index{Robidoux, Nicolas}
+\begin{chunk}{axiom.bib}
+@misc{Robi93,
+ author = "Robidoux, Nicolas",
+ title = "Does Axiom Solve Systems of O.D.E's Like Mathematica?",
+ year = "1993",
+ paper = "Robi93.pdf",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/Robidoux.pdf",
+ keywords = "axiomref",
+ abstract = "
+ If I were demonstrating Axiom and were asked this question, my reply
+ would be ``No, but I am not sure that this is a bad thing''. And I
+ would illustrate this with the following example.
+
+ Consider the following system of O.D.E.'s
+ \[
+ \begin{array}{rcl}
+ \frac{dx_1}{dt} & = & \left(1+\frac{cos t}{2+sin t}\right)x_1\\
+ \frac{dx_2}{dt} & = & x_1  x_2
+ \end{array}
+ \]
+ This is a very simple system: $x_1$ is actually uncoupled from $x_2$"
+}
+
+\end{chunk}
+
+\index{Davenport, James H.}
+\index{Faure, Christ\'ele}
+\begin{chunk}{axiom.bib}
+@misc{Davexx,
+ author = {Davenport, James; Faure, Christ\'ele},
+ title = "The Unknown in Computer Algebra",
+ url =
+"http://axiomwiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf",
+ paper = "Davexx.pdf",
+ keywords = "axiomref",
+ abstract = "
+ Computer algebra systems have to deal with the confusion between
+ ``programming variables'' and ``mathematical symbols''. We claim that
+ they should also deal with ``unknowns'', i.e. elements whose values
+ are unknown, but whose type is known. For examples $x^p \ne x$ if $x$
+ is a symbol, but $x^p = x$ if $x \in GF(p)$. We show how we have
+ extended Axiom to deal with this concept."
+}
+
+\end{chunk}
+
+\index{Davenport, James H.}
+\begin{chunk}{axiom.bib}
+@techreport{Dave92b,
+ author = "Davenport, James H.",
+ title = "How does one program in the AXIOM system?",
+ institution = "Numerical Algorithms Group, Inc.",
+ year = "1992",
+ type = "technical report",
+ number = "TR6/92 (ATR/4)(NP2493)",
+ url = "http://www.nag.co.uk/doc/TechRep/axiomtr.html",
+ paper = "Dave92b.pdf",
+ keywords = "axiomref",
+ abstract =
+ "Axiom is a computer algebra system superficially like many others, but
+ fundamentally different in its internal construction, and therefore in
+ the possibilities it offers to its users and programmers. In these
+ lecture notes, we will explain, by example, the methodology that the
+ author uses for programming substantial bits of mathematics in Axiom."
+}
+
+\end{chunk}
+
+\index{Youssef, Saul}
+\begin{chunk}{axiom.bib}
+@misc{Yous04,
+ author = "Youssef, Saul",
+ title = "Prospects for Category Theory in Aldor",
+ year = "2004",
+ url =
+"http://axiomwiki.newsynthesis.org/public/refs/YoussefProspectsForCategoryTheoryInAldor.pdf",
+ paper = "Yous04.pdf",
+ abstract =
+ "Ways of encorporating category theory constructions and results into
+ the Aldor language are discussed. The main features of Aldor which
+ make this possible are identified, examples of categorical
+ constructions are provided and a suggestion is made for a foundation
+ for rigorous results."
+}
+
+\end{chunk}
+
+\index{Carpent, Quentin}
+\index{Conil, Christophe}
+\begin{chunk}{axiom.bib}
+@misc{Carp04,
+ author = "Carpent, Quentin and Conil, Christophe",
+ title = "Utilisation de logiciels libres pour la r\'ealisation de TP MT26",
+ year = "2004",
+ paper = "Carp04.pdf",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/ac20.pdf",
+ keywords = "axiomref",
+ comment = "french",
+ abstract = "radicalSolve(x**3+x**27=0,x)"
+}
+
+\end{chunk}
+
+\index{Naylor, William A.}
+\index{Padget, Julian}
+\begin{chunk}{axiom.bib}
+@InProceedings{Nayl06,
+ author = "Naylor, William and Padget, Julian",
+ title = "From Untyped to Polymorphically Typed Objects in Mathematical
+ Web Services",
+ paper = "NPxx.pdf",
+ series = Lecture Notes in Computer Science",
+ volume = "4108",
+ pages = "222236",
+ year = "2006",
+ keywords = "axiomref",
+ abstract =
+ "OpenMath is a widely recognized approach to the semantic markup of
+ mathematics that is often used for communication between OpenMath
+ compliant systems. The Aldor language has a sophisticated
+ categorybased type system that was specifically developed for the
+ purpose of modelling mathematical structures, while the system itself
+ supports the creation of smallfootprint applications suitable for
+ deployment as web services. In this paper we present our first results
+ of how one may perform translations from generic OpenMath objects into
+ values in specific Aldor domains, describing how the Aldor interfae
+ domain ExpresstionTree is used to achieve this. We outline our Aldor
+ implementation of an OpenMath translator, and describe an efficient
+ extention of this to the Parser category. In addition, the Aldor
+ service creation and invocation mechanism are explained. Thus we are
+ in a position to develop and deploy mathematical web services whose
+ descriptions may be directly derived from Aldor's rich type language."
+}
+
+\end{chunk}
+
+\index{Watt, Stephen M.}
+\index{Broadbery, Peter A.}
+\index{Dooley, Sam}
+\index{Iglio, Pietro}
+\begin{chunk}{axiom.bib}
+@techreport{Watt94,
+ author = "Watt, Stephen M. and Broadbery, Peter A. and Dooley, Samuel S.
+ and Iglio, Pietro",
+ title = "A First Report on the A\# Compiler (including benchmarks)",
+ institution = "IBM Research",
+ year = "1994",
+ type = "technical report",
+ number = "RC19529 (85075)",
+ paper = "Watt94.pdf",
+ url =
+ "http://axiomwiki.newsynthesis.org/public/refs/axiomaldorasharp.pdf",
+ keywords = "axiomref",
+ abstract =
+ "The $A^{#}$ compiler allows users of computer algebra to develop
+ programs in a context where multiple programming languages are
+ employed. The compiler translates programs written in the $A^{#}$
+ programming language to a lowlevel intermediate language, Foam,
+ from which it can generate standalone programs, native object
+ libraries to be linked with other applications, or code to be read
+ into closed environments. In addition, Foam code may be directly
+ executed using an interpreter provided with the $A^{#}$ compiler.
+
+ The $A^{#}$ programming language provides support for objectoriented
+ and functional programming styles. It is ``higherorder'' in the sense
+ that both types and functions are first class, and may be manipulated
+ in the same ways as any other values. The primary considerations in
+ the formulation of the language have been generality, composibility,
+ and efficiency. The language has been designed to admit a number of
+ important optimizations, allowing compilation to machine code which is
+ in many instances of efficiency comparable to that produced by a C or
+ Fortran compiler.
+
+ The original motivation for $A^{#}$ comes from the field of computer
+ algebra: to provide an improved extension language for the Axiom
+ computer algebra system."
+}
+
+\end{chunk}
+
+\index{Lambe, Larry A.}
+\index{Luczak, Richard}
+\begin{chunk}{axiom.bib}
+@article{Lamb93a,
+ article = "Lambe, Larry and Luczak, Richard",
+ title = "ObjectOriented Mathematical Programming and
+ Symbolic/Numeric Interface",
+ journal = "3rd Int. Conf. on Expert Systems in Numerical Computing",
+ year = "1993",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/axiomfem.pdf",
+ paper = "Lamb93a.pdf",
+ keywords = "axiomref",
+ abstract =
+ "The Axiom language is based on the notions of ``categories'',
+ ``domains'', and ``packages''. These concepts are used to build an
+ interface between symbolic and numeric calculations. In particular, an
+ interface to the NAG Fortran Library and Axiom's algebra and graphics
+ facilities is presented. Some examples of numerical calculations in a
+ symbolic computational environment are also included using the finite
+ element method. While the examples are elementary, we believe that
+ they point to very powerful methods for combining numeric and symbolic
+ computational techniques."
+}
+
+\end{chunk}
+
+\index{Griesmer, James H.}
+\index{Jenks, Richard D.}
+\begin{chunk}{axiom.bib}
+@InProceedings{Grie71,
+ author = "Griesmer, James H. and Jenks, Richard D.",
+ title = "SCRATCHPAD/1  an interactive facility for symbolic mathematics",
+ booktitle = "Proc. second ACM Symposium on Symbolic and Algebraic
+ Manipulation",
+ series = "SYMSAC 71",
+ year = "1971",
+ pages = "4258",
+ url = "http://delivery.acm.org/10.1145/810000/806266/p42griesmer.pdf",
+ paper = "GJ71.pdf",
+ keywords = "axiomref",
+ abstract = "
+ The SCRATCHPAD/1 system is designed to provide an interactive symbolic
+ computational facility for the mathematician user. The system features
+ a user language designed to capture the style and succinctness of
+ mathematical notation, together with a facility for conveniently
+ introducing new notations into the language. A comprehensive system
+ library incorporates symbolic capabilities provided by such systems as
+ SIN, MATHLAB, and REDUCE."
+}
+
+\end{chunk}
+
+\index{Seiler, Werner Markus}
+\index{Calmet, J.}
+\begin{chunk}{axiom.bib}
+@misc{Seil95a,
+ author = "Seiler, Werner Markus and Calmet, J.",
+ title = "JET  An Axiom Environment for Geometric Computations with
+ Differential Equations",
+ paper = "Seil95a.pdf",
+ url = "http://axiomwiki.newsynthesis.org/public/refs/axiomjet95.pdf",
+ keywords = "axiomref",
+ abstract =
+ "JET is an environment within the computer algebra system Axiom to
+ perform such computations. The current implementation emphasises the
+ two key concepts involution and symmetry. It provides some packages
+ for the completion of a given system of differential equations to an
+ equivalent involutive one based on the CartanKuranishi theorem and
+ for setting up the determining equations for classical and
+ nonclassical point symmetries."
+}
+
+\end{chunk}
Somewhere along the way I fatfingered a character delete
causing the build to break. Sigh.
diff git a/src/axiomwebsite/patches.html b/src/axiomwebsite/patches.html
index e23acf0..4c07d74 100644
 a/src/axiomwebsite/patches.html
+++ b/src/axiomwebsite/patches.html
@@ 5424,6 +5424,8 @@ books/bookvolbib Axiom Citations in the Literature
books/bookvolbib Axiom Citations in the Literature
20160628.02.tpd.patch
src/input/Makefile fix typo
+20160629.01.tpd.patch
+books/bookvolbib Axiom Citations in the Literature

1.7.5.4