diff git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet
index 8549560..eb8d61f 100644
 a/books/bookvolbib.pamphlet
+++ b/books/bookvolbib.pamphlet
@@ 10,10695 +10,10324 @@ initially derived with permission from Nelson Beebe's collection.
The second section contains references from Axiom to the literature.
The third section sorts papers by topic.
\chapter{The Bibliography}
\section{Axiom Citations in the Literature}

\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[ACM 89]{ACM89} ACM, editor
Proceedings of the ACMSIGSAM 1989 International
Symposium on Symbolic and Algebraic Computation, ISSAC '89 ACM Press,
New York, NY 10036, USA, 1989, , LCCN QA76.95.I59
 year = "1989",
 isbn = "0897913256",
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[ACM 94]{ACM94} ACM, editor
ISSAC '94. Proceedings of the International
Symposium on Symbolic and Algebraic Computation. ACM Press, New York, NY,
10036, USA, 1994, . LCCN QA76.95.I59
 year = "1994",
 isbn = "0897916387",
 keywords = "axiomref",
+\section{Special Topics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{chunk}
+\subsection{Solving Systems of Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@article{Augo91,
 author = "Augot, D. and Charpin, P. and Sendrier, N.",
 title = "The miniumum distance of some binary codes via the Newton's identities",
 journal = "Cohen and Charping [CC91]",
 year = "1991",
 pages = "6573",
 isbn = "0387543031",
 misc = "3540543031 (Berlin). LCCN QA268.E95 1990",
 keywords = "axiomref",
 paper = "Augo91.pdf"
+@inproceedings{Bro86,
+ author = "Bronstein, Manuel",
+ title = "Gsolve: a faster algorithm for solving systems of algebraic
+ equations",
+ booktitle = "Proc of 5th ACM SYMSAC",
+ year = "1986",
+ pages = "247249",
+ isbn = "0897911997",
+ abstract = "
+ We apply the elimination property of Gr{\"o}bner bases with respect to
+ pure lexicographic ordering to solve systems of algebraic equations.
+ We suggest reasons for this approach to be faster than the resultant
+ technique, and give examples and timings that show that it is indeed
+ faster and more correct, than MACSYMA's solve."
}
\end{chunk}
+\subsection{Numerical Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Adams 94]{AL94}
 author = "Adams, William W. and Loustaunau, Philippe",
 title = "An Introduction to Gr\"obner Bases",
 year = "1994",
American Mathematical Society (1994)
 isbn = "0821838040",
 keywords = "axiomref",
+{Bro99,
+ author = "Bronstein, Manuel",
+ title = "Fast Deterministic Computation of Determinants of Dense Matrices",
+ url = "http://wwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html",
+ paper = "Bro99.pdf",
+ abstract = "
+ In this paper we consider deterministic computation of the exact
+ determinant of a dense matrix $M$ of integers. We present a new
+ algorithm with worst case complexity
+ \[O(n^4(log n+ log \verb?M?)+x^3 log^2 \verb?M?)\],
+ where $n$ is the dimension of the matrix
+ and \verb?M? is a bound on the entries in $M$, but with
+ average expected complexity
+ \[O(n^4+m^3(log n + log \verb?M?)^2)\],
+ assuming some plausible properties about the distribution of $M$.
+ We will also describe a practical version of the algorithm and include
+ timing data to compare this algorithm with existing ones. Our result
+ does not depend on ``fast'' integer or matrix techniques."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Andrews 84]{And84}
 author = "Andrews, George E.",
 title = "Ramanujan and SCRATCHPAD",
 year = "1984",
 pages = "383??",
 keywords = "axiomref",
In Golden and Hussain [GH84]
+{Kel00,
+ author = "Kelsey, Tom",
+ title = "Exact Numerical Computation via Symbolic Computation",
+ url = "http://tom.host.cs.standrews.ac.uk/pub/ccapaper.pdf",
+ paper = "Kel00.pdf",
+ abstract = "
+ We provide a method for converting any symbolic algebraic expression
+ that can be converted into a floating point number into an exact
+ numeric representation. We use this method to demonstrate a suite of
+ procedures for the representation of, and arithmetic over, exact real
+ numbers in the Maple computer algebra system. Exact reals are
+ represented by potentially infinite lists of binary digits, and
+ interpreted as sums of negative powers of the golden ratio."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Andrews 88]{And88}
 author = "Andrews, G. E.",
 title = "Application of Scratchpad to problems in special functions and combinatorics",
 year = "1988"
 pages = "158??",
 isbn = "3540189289",
 keywords = "axiomref",
In Janssen [Jan88], pages 158?? ISBN
0387189289 LCCN QA155.7.E4T74
+{Yang14,
+ author ="Yang, Xiang and Mittal, Rajat",
+ title = "Acceleration of the Jacobi iterative method by factors exceeding 100
+ using scheduled relation",
+ url =
+"http://engineering.jhu.edu/fsag/wpcontent/uploads/sites/23/2013/10/JCP_revised_WebPost.pdf",
+ paper = "Yang14.pdf"
+}
\end{chunk}
+\subsection{Special Functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Anon 91]{Ano91}
 author = "Anonymous",
 year = "1991,
 keywords = "axiomref",
Proceedings 1991 Annual Conference, American Society for
Engineering Education. Challenges of a Changing World. ASEE, Washington, DC
 2 vol.
+{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}
\begin{chunk}{ignore}
\bibitem[Anon 92]{Ano92}
 author = "Anonymous",
 year = "1992",
 keywords = "axiomref",
Programming environments for highlevel scientific problem solving.
IFIP TC2/WG 2.5 working conference. IFIP Transactions. A Computer Science
and Technology, A2:??, CODEN ITATEC. ISSN 09265473
+{Ng68,
+ author = "Ng, Edward W. and Geller, Murray",
+ title = "A Table of Integrals of the Error functions",
+ url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf",
+ paper = "Ng68.pdf",
+ abstract = "
+ This is a compendium of indefinite and definite integrals of products
+ of the Error functions with elementary and transcendental functions."
+}
\end{chunk}
+\subsection{Exponential Integral $E_1(x)$} %%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Anono 95]{Ano95}
 author =Anonymous
 keywords = "axiomref",
 year = "1995",
GAMM 94 annual meeting. Zeitschrift fur Angewandte Mathematik und
Physik, 75 (suppl. 2), CODEN ZAMMAX, ISSN 00442267
+{Gell69,
+ author = "Geller, Murray and Ng, Edward W.",
+ title = "A Table of Integrals of the Exponential Integral",
+ url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf",
+ paper = "Gell69.pdf",
+ abstract = "
+ This is a compendium of indefinite and definite integrals of products
+ of the Exponential Integral with elementary or transcendental functions."
+}
\end{chunk}
\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{axiom.bib}
@article{Bacl14,
 author = "Baclawski, Krystian",
 title = "SPAD language type checker",
 journal = "unknown",
 year = "2014",
 url = "http://github.com/cahirwpz/phd",
 keywords = "axiomref"
+@techreport{Segl98,
+ author = "Segletes, S.B.",
+ title = "A compact analytical fit to the exponential integral $E_1(x)$",
+ year = "1998",
+ institution = "U.S. Army Ballistic Research Laboratory,
+ Aberdeen Proving Ground, MD",
+ type = "Technical Report",
+ number = "ARLTR1758",
+ paper = "Segl98.pdf",
+ abstract = "
+ A fourparameter fit is developed for the class of integrals known as
+ the exponential integral (real branch). Unlike other fits that are
+ piecewise in nature, the current fit to the exponential integral is
+ valid over the complete domain of the function (compact) and is
+ everywhere accurate to within $\pm 0.0052\%$ when evaluating the first
+ exponential integral, $E_1$. To achieve this result, a methodology
+ that makes use of analytically known limiting behaviors at either
+ extreme of the domain is employed. Because the fit accurately captures
+ limiting behaviors of the $E_1$ function, more accuracy is retained
+ when the fit is used as part of the scheme to evaluate higherorder
+ exponential integrals, $E_n$, as compared with the use of bruteforce
+ fits to $E_1$, which fail to accurately model limiting
+ behaviors. Furthermore, because the fit is compact, no special
+ accommodations are required (as in the case of spliced piecewise fits)
+ to smooth the value, slope, and higher derivatives in the transition
+ region between two piecewise domains. The general methodology employed
+ to develop this fit is outlined, since it may be used for other
+ problems as well."
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@techreport{Se09,
+ author = "Segletes, S.B.",
+ title = "Improved fits for $E_1(x)$ {\sl vis\'avis} those presented
+ in ARLTR1758",
+ type = "Technical Report",
+ number = "ARLTR1758",
+ institution ="U.S. Army Ballistic Research Laboratory,
+ Aberdeen Proving Ground, MD",
+ year = "1998",
+ month = "September",
+ paper = "Se09.pdf",
+ abstract = "
+ This is a writeup detailing the more accurate fits to $E_1(x)$,
+ relative to those presented in ARLTR1758. My actual fits are to
+ \[F1 =[x\ exp(x) E_1(x)]\] which spans a functional range from 0 to 1.
+ The best accuracy I have been yet able to achieve, defined by limiting
+ the value of \[[(F1)_{fit}  F1]/F1\] over the domain, is
+ approximately 3.1E07 with a 12parameter fit, which unfortunately
+ isn't quite to 32bit floatingpoint accuracy. Nonetheless, the fit
+ is not a piecewise fit, but rather a single continuous function over
+ the domain of nonnegative x, which avoids some of the problems
+ associated with piecewise domain splicing."
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The project aims to deliver a new type checker for SPAD language.
Several improvements over current type checker are planned.
\begin{itemize}
\item introduce better type inference
\item introduce modern language constructs
\item produce understandable diagnostic messages
\item eliminate well known bugs in the type system
\item find new type errors
\end{itemize}
\end{adjustwidth}
+\subsection{Polynomial GCD} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Blair 70]{BGJ70}
 author = "Blair, Fred W and Griesmer, James H. and Jenks, Richard D.",
 title = "An interactive facility for symbolic mathematics",
 year = "1970",
 pages = "394419",
 keywords = "axiomref",
Proc. International Computing Symposium, Bonn, Germany,
+\bibitem[Knuth 71]{STPGCDKnu71} Knuth, Donald
+``The Art of Computer Programming''
+2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing,
+AddisonWesley 1971, section 4.6 pp399505
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Blair 70a]{BJ70}
 author = "Blair, Fred W. and Jenks, Richard D.",
 title = "LPL: LISP programming language",
 year = "1970",
 keywords = "axiomref",
IBM Research Report, RC3062 Sept
+\bibitem[Ma 90]{STPGCDMa90} Ma, Keju; Gathen, Joachim von zur
+``Analysis of Euclidean Algorithms for Polynomials over Finite Fields''
+J. Symbolic Computation (1990) Vol 9 pp429455\hfill{}
+\verbwww.researchgate.net/publication/220161718_Analysis_of_Euclidean_
+\verbAlgorithms_for_Polynomials_over_Finite_Fields/file/
+\verb60b7d52b326a1058e4.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/STPGCDMa90.pdf
+ abstract = "
+ This paper analyzes the Euclidean algorithm and some variants of it
+ for computing the greatest common divisor of two univariate polynomials
+ over a finite field. The minimum, maximum, and average number of
+ arithmetic operations both on polynomials and in the ground field
+ are derived."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Broadbery 95]{BGDW95}
 author = "Broadbery, P. A. and G{\'o}mezD{\'\i}az, T. and Watt, S. M.",
 title = "On the Implementation of Dynamic Evaluation",
 year = "1995",
 pages = "7784",
 keywords = "axiomref",
 isbn = "0897916999",
 url = "http://pdf.aminer.org/000/449/014/on_the_implementation_of_dynamic_evaluation.pdf",
 paper = "BGDW95.pdf"
In Levelt [Lev95] 0897916999 LCCN QA76.95 I59 1995
ACM order number 505950
+\bibitem[Naylor 00a]{N00} Naylor, Bill
+``Polynomial GCD Using Straight Line Program Representation''
+PhD. Thesis, University of Bath, 2000
+\verbwww.sci.csd.uwo.ca/~bill/thesis.ps
+%\verbaxiomdeveloper.org/axiomwebsite/papers/N00.pdf
+ abstract = "
+ This thesis is concerned with calculating polynomial greatest common
+ divisors using straight line program representation.
+
+ In the Introduction chapter, we introduce the problem and describe
+ some of the traditional representations for polynomials, we then talk
+ about some of the general subjects central to the thesis, terminating
+ with a synopsis of the category theory which is central to the Axiom
+ computer algebra system used during this research.
+
+ The second chapter is devoted to describing category theory. We follow
+ with a chapter detailing the important sections of computer code
+ written in order to investigate the straight line program subject.
+ The following chapter on evalution strategies and algorithms which are
+ dependant on these follows, the major algorith which is dependant on
+ evaluation and which is central to our theis being that of equality
+ checking. This is indeed central to many mathematical problems.
+ Interpolation, that is the determination of coefficients of a
+ polynomial is the subject of the next chapter. This is very important
+ for many straight line program algorithms, as their noncanonical
+ structure implies that it is relatively difficult to determine
+ coefficients, these being the basic objects that many algorithms work
+ on. We talk about three separate interpolation techniques and compare
+ their advantages and disadvantages. The final two chapters describe
+ some of the results we have obtained from this research and finally
+ conclusions we have drawn as to the viability of the straight line
+ program approach and possible extensions.
+
+ Finally we terminate with a number of appendices discussing side
+ subjects encountered during the thesis."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Dynamic evaluation is a technique for producing multiple results
according to a decision tree which evolves with program execution.
Sometimes it is desired to produce results for all possible branches
in the decision tree, while on other occasions, it may be sufficient
to compute a single result which satisfies certain properties. This
techinique finds use in computer algebra where computing the correct
result depends on recognizing and properly handling special cases of
parameters. In previous work, programs using dynamic evaluation have
explored all branches of decision trees by repeating the computations
prior to decision points.

This paper presents two new implementations of dynamic evaluation
which avoid recomputing intermediate results. The first approach uses
Scheme ``continuations'' to record state for resuming program
execution. The second implementation uses the Unix ``fork'' operation
to form new processes to explore alternative branches in parallel.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Boehm 89]{Boe89}
 author = "Boehm, HansJ.",
 title = "Type Inference in the Presence of Type Abstraction",
 year = "1989",
 pages = "192206",
 keywords = "axiomref",
 url = "http://www.acm.org/pubs/citations/proceedings/pldi/73141/p192boehm",
 paper = "Boe89.pdf",
ACM SIGPLAN Notices, 24(7) pp July CODEN SINODQ ISSN 03621340
+\bibitem[Shoup 93]{STPGCDSh93} Shoup, Victor
+``Factoring Polynomials over Finite Fields: Asymptotic Complexity vs
+Reality*''
+Proc. IMACS Symposium, Lille, France, (1993)
+\verbwww.shoup.net/papers/lille.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/STPGCDSh93.pdf
+ abstract = "
+ This paper compares the algorithms by Berlekamp, Cantor and
+ Zassenhaus, and Gathen and Shoup to conclude that (a) if large
+ polynomials are factored the FFT should be used for polynomial
+ multiplication and division, (b) Gathen and Shoup should be used if
+ the number of irreducible factors of $f$ is small. (c) if nothing is
+ know about the degrees of the factors then Berlekamp's algorithm
+ should be used."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A number of recent programming language designs incorporate a type
checking system based on the GirardReynolds polymorphic
$\lambda$calculus. This allows the construction of general purpose,
reusable software without sacrificing compiletime type checking. A
major factor constraining the implementation of these languages is the
difficulty of automatically inferring the lengthy type information
that is otherwise required if full use is made of these
languages. There is no known algorithm to solve any natural and fully
general formulation of the ``type inference'' problem. One very
reasonable formulation of the problem is known to be undecidable.

Here we define a restricted version of the type inference problem and
present an efficient algorithm for its solution. We argue that the
restriction is sufficiently weak to be unobtrusive in practice.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Boulton 04]{BHGM04}
 author = "Boulton, Richard and Hardy, Ruth and Gottliebsen, Hanne and Martin, Ursula",
 title = "Design verification for control engineering",
 year = "2004",
 month = "April",
Proc Fourth International Conference on Integrated Formal Methods,
 keywords = "axiomref",
+\bibitem[Gathen 01]{STPGCDGa01} Gathen, Joachim von zur; Panario, Daniel
+``Factoring Polynomials Over Finite Fields: A Survey''
+J. Symbolic Computation (2001) Vol 31, pp317\hfill{}
+\verbpeople.csail.mit.edu/dmoshdov/courses/codes/polyfactorization.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/STPGCDGa01.pdf
+ keywords = "survey",
+ abstract = "
+ This survey reviews several algorithms for the factorization of
+ univariate polynomials over finite fields. We emphasize the main ideas
+ of the methods and provide and uptodate bibliography of the problem.
+ This paper gives algorithms for {\sl squarefree factorization},
+ {\sl distinctdegree factorization}, and {\sl equaldegree factorization}.
+ The first and second algorithms are deterministic, the third is
+ probabilistic."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We introduce control engineering as a new domain of application for
formal methods. We discuss design verification, drawing attention to
the role played by diagrammatic evaluation criteria involving numeric
plots of a design, such as Nichols and Bode plots. We show that
symbolic computation and computational logic can be used to discharge
these criteria and provide symbolic, automated, and very general
alternatives to these standard numeric tests. We illustrate our work
with reference to a standard reference model drawn from military
avionics.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Boulanger 91]{Bou91}
 author = "Boulanger, JeanLouis",
 title = "Etude de la compilation de scratchpad 2",
 year = "1991",
 month = "September",
Rapport de DEA Universite dl lille 1
 keywords = "axiomref",
+\bibitem[van Hoeij]{Hoeij04} Hoeij, Mark van; Monagen, Michael
+``Algorithms for Polynomial GCD Computation over Algebraic Function Fields''
+\verbwww.cecm.sfu.ca/personal/mmonagan/papers/AFGCD.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Hoeij04.pdf
+ abstract = "
+ Let $L$ be an algebraic function field in $k \ge 0$ parameters
+ $t_1,\ldots,t)k$. Let $f_1$, $f_2$ be nonzero polynomials in
+ $L[x]$. We give two algorithms for computing their gcd. The first, a
+ modular GCD algorithm, is an extension of the modular GCD algorithm
+ for Brown for {\bf Z}$[x_1,\ldots,x_n]$ and Encarnacion for {\bf
+ Q}$(\alpha[x])$ to function fields. The second, a fractionfree
+ algorithm, is a modification of the Moreno Maza and Rioboo algorithm
+ for computing gcds over triangular sets. The modification reduces
+ coefficient grownth in $L$ to be linear. We give an empirical
+ comparison of the two algorithms using implementations in Maple."
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Bou93a,
 author = "Boulanger, JeanLouis",
 title = "Axiom, language fonctionnel \`a d\'evelopement objet",
 year = "1993",
 month = "October",
 paper = "Bou93a.pdf",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Wang 78]{Wang78} Wang, Paul S.
+``An Improved Multivariate Polynomial Factoring Algorithm''
+Mathematics of Computation, Vol 32, No 144 Oct 1978, pp12151231
+\verbwww.ams.org/journals/mcom/197832144/S00255718197805682843/
+\verbS00255718197805682843.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Wang78.pdf
+ abstract = "
+ A new algorithm for factoring multivariate polynomials over the
+ integers based on an algorithm by Wang and Rothschild is described.
+ The new algorithm has improved strategies for dealing with the known
+ problems of the original algorithm, namely, the leading coefficient
+ problem, the badzero problem and the occurence of extraneous factors.
+ It has an algorithm for correctly predetermining leading coefficients
+ of the factors. A new and efficient padic algorith named EEZ is
+ described. Basically it is a linearly convergent variablebyvariable
+ parallel construction. The improved algorithm is generally faster and
+ requires less store than the original algorithm. Machine examples with
+ comparative timing are included."
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Bou93b,
 author = "Boulanger, JeanLouis",
 title = "AXIOM, A Functional Language with Object Oriented Development",
 year = "1993",
 paper = "Bou93b.pdf",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Wiki 4]{Wiki4}.
+``Polynomial greatest common divisor''
+\verben.wikipedia.org/wiki/Polynomial_greatest_common_divisor
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present in this paper, a study about the computer algebra system
Axiom, which gives us many very interesting Software engineering
concepts. This language is a functional language with an Object
Oriented Development. This feature is very important for modeling the
mathematical world (Hierarchy) and provides a running with
mathematical sense. (All objects are functions). We present many
problems of running and development in Axiom. We can note that Aiom is
the only system of this category.
\end{adjustwidth}
+\subsection{Category Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Boulanger 94]{Bou94}
 author = "Boulanger, J.L.",
 title = "Object Oriented Method for Axiom",
 year = "1995",
 month = "February",
 pages = "3341",
 paper = "Bou94.pdf",
ACM SIGPLAN Notices, 30(2) CODEN SINODQ ISSN 03621340
 keywords = "axiomref",
+\bibitem[Baez 09]{Baez09} Baez, John C.; Stay, Mike
+``Physics, Topology, Logic and Computation: A Rosetta Stone''
+\verbarxiv.org/pdf/0903.0340v3.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Baez09.pdf
+ abstract = "
+ In physics, Feynman diagrams are used to reason about quantum
+ processes. In the 1980s, it became clear that underlying these
+ diagrams is a powerful analogy between quantum physics and
+ topology. Namely, a linear operator behaves very much like a
+ ``cobordism'': a manifold representing spacetime, going between two
+ manifolds representing space. But this was just the beginning: simiar
+ diagrams can be used to reason about logic, where they represent
+ proofs, and computation, where they represent programs. With the rise
+ of interest in quantum cryptography and quantum computation, it became
+ clear that there is an extensive network of analogies between physics,
+ topology, logic and computation. In this expository paper, we make
+ some of these analogies precise using the concept of ``closed
+ symmetric monodial category''. We assume no prior knowledge of
+ category theory, proof theory or computer science."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Axiom is a very powerful computer algebra system which combines two
language paradigms (functional and OOP). Mathematical world is complex
and mathematicians use abstraction to design it. This paper presents
some aspects of the object oriented development in Axiom. The Axiom
programming is based on several new tools for object oriented
development, it uses two levels of class and some operations such that
{\sl coerce}, {\sl retract}, or {\sl convert} which permit the type
evolution. These notions introduce the concept of multiview.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 87]{Bro87}
 author = "Bronstein, Manuel",
 title = "Integration of Algebraic and Mixed Functions",
 year = "1987",
in [Wit87], p18
 keywords = "axiomref",
+\bibitem[Meijer 91]{Meij91} Meijer, Erik; Fokkinga, Maarten; Paterson, Ross
+``Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire''
+\verbeprints.eemcs.utwente.nl/7281/01/dbutwente40501F46.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/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}
\begin{chunk}{ignore}
\bibitem[Bronstein 89]{Bro89}
 author= "Bronstein, M.",
 title = "Simplification of real elementary functions",
 year = "1989",
 pages = "207211",
 isbn = "0897913256",
ACM [ACM89] pages LCCN QA76.95.I59 1989
 keywords = "axiomref",
+\bibitem[Youssef 04]{You04} Youssef, Saul
+``Prospects for Category Theory in Aldor''
+October 2004
+%\verbaxiomdeveloper.org/axiomwebsite/papers/You04.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}
\begin{adjustwidth}{2.5em}{0pt}
We describe an algorithm, based on Risch's real structure theorem, that
determines explicitly all the algebraic relations among a given set of
real elementary functions. We also provide examples from its
implementation that illustrate the advantages over the use of complex
logarithms and exponentials.
\end{adjustwidth}
+\subsection{Proving Axiom Correct} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
\bibitem[Bronstein 91a]{Bro91a}
@inproceedings{Bron91a,
 author = "Bronstein, M.",
 title = "The Risch Differential Equation on an Algebraic Curve",
 booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation",
 series = "ISSAC'91",
 year = "1991",
 pages = "241246",
 isbn = "0897914376",
 publisher = "ACM, NY",
 keywords = "axiomref",
 paper = "Bro91a.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Adams 99]{Adam99} Adams, A.A.; Gottlieben, H.; Linton, S.A.;
+Martin, U.
+``Automated theorem proving in support of computer algebra:''
+`` symbolic definite integration as a case study''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Adam99.pdf
+ abstract = "
+ We assess the current state of research in the application of computer
+ aided formal reasoning to computer algebra, and argue that embedded
+ verification support allows users to enjoy its benefits without
+ wrestling with technicalities. We illustrate this claim by considering
+ symbolic definite integration, and present a verifiable symbolic
+ definite integral table look up: a system which matches a query
+ comprising a definite integral with parameters and side conditions,
+ against an entry in a verifiable table and uses a call to a library of
+ lemmas about the reals in the theorem prover PVS to aid in the
+ transformation of the table entry into an answer. We present the full
+ model of such a system as well as a description of our prototype
+ implementation showing the efficacy of such a system: for example, the
+ prototype is able to obtain correct answers in cases where computer
+ algebra systems [CAS] do not. We extend upon Fateman's webbased table
+ by including parametric limits of integration and queries with side
+ conditions."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present a new rational algorithm for solving Risch differential
equations over algebraic curves. This algorithm can also be used to
solve $n^{th}$order linear ordinary differential equations with
coefficients in an algebraic extension of the rational functions. In
the general ("mixed function") case, this algorithm finds the
denominator of any solution of the equation.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 91c]{Bro91c}
 author = "Bronstein, Manuel",
 title = "Computer Algebra and Indefinite Integrals",
 year = "1991",
 paper = "Bro91c.pdf",
in Computer Aided Proofs in Analysis, K.R. Meyers et al. (eds)
SpringerVerlag, NY (1991)
 keywords = "axiomref",
+\bibitem[Adams 01]{Adam01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne;
+Kelsey, Tom; Martin, Ursula; Owre, Sam
+``Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS''
+\verbwww.csl.sri.com/~owre/papers/tphols01/tphols01.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Adam01.pdf
+ abstract = "
+ We describe an interface between version 6 of the Maple computer
+ algebra system with the PVS automated theorem prover. The interface is
+ designed to allow Maple users access to the robust and checkable proof
+ environment of PVS. We also extend this environment by the provision
+ of a library of proof strategies for use in real analysis. We
+ demonstrate examples using the interface and the real analysis
+ library. These examples provide proofs which are both illustrative and
+ applicable to genuine symbolic computation problems."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We give an overview, from an analytical point of view, of decision
procedures for determining whether an elementary function has an
elementary function has an elementary antiderivative. We give examples
of algebraic functions which are integrable and nonintegrable in
closed form, and mention the current implementation of various computer
algebra systems.
\end{adjustwidth}
+\begin{chunk}{axiom.bib}
+@article{Mahb06,
+ author = "Mahboubi, Assia",
+ title = "Proving Formally the Implementation of an Efficient gcd
+ Algorithm for Polynomials",
+ journal = "Lecture Notes in Computer Science",
+ volume = "4130",
+ year = "2006",
+ pages = "438452",
+ paper = "Mahb06.pdf",
+ abstract = "
+ We describe here a formal proof in the Coq system of the structure
+ theorem for subresultants which allows to prove formally the
+ correctness of our implementation of the subresultants algorithm.
+ Up to our knowledge it is the first mechanized proof of this result."
+}
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 92]{Bro92}
 author = "Bronstein, M.",
 title = "Linear Ordinary Differential Equations: Breaking Through the Order 2 Barrier",
 year = "1992",
 url = "http://wwwsop.inria.fr/cafe/Manuel.Bronstein/publications/issac92.ps.gz",
 paper = "Bro92.pdf",
 keywords = "axiomref",
+\bibitem[Ballarin 99]{Ball99} Ballarin, Clemens; Paulson, Lawrence C.
+``A Pragmatic Approach to Extending Provers by Computer Algebra 
+ with Applications to Coding Theory''
+\verbwww.cl.cam.ac.uk/~lp15/papers/Isabelle/coding.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Ball99.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 proofs depend 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 formal specifications for the algorithms, and links the
+ computer algebra library Sumit to the prover Isabelle.
+
+ We give details of the interface, the use of the computer algebra
+ system on the tacticlevel of Isabelle and its integration into proof
+ procedures."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A major subproblem for algorithms that either factor ordinary linear
differential equations or compute their closed form solutions is to
find their solutions $y$ which satisfy $y^{'}/y \in \overline{K}(x)$
where $K$ is the constant field for the coefficients of the equation.
While a decision procedure for this subproblem was known in the
$19^{th}$ century, it requires factoring polynomials over
$\overline{K}$ and has not been implemented in full generality. We
present here an efficient algorithm for this subproblem, which has
been implemented in the AXIOM computer algebra system for equations of
arbitrary order over arbitrary fields of characteristic 0. This
algorithm never needs to compute with the individual complex
singularities of the equation, and algebraic numbers are added only
when they appear in the potential solutions. Implementation of the
complete Singer algorithm for $n=2,3$ based on this building block is
in progress.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Bertot 04]{Bert04} Bertot, Yves; Cast\'eran, Pierre
+``Interactive Theorem Proving and Program Development''
+Springer ISBN 3540208542
+ abstract = "
+ Coq is an interactive proof assistant for the development of
+ mathematical theories and formally certified software. It is based on
+ a theory called the calculus of inductive constructions, a variant of
+ type theory.
+
+ This book provides a pragmatic introduction to the development of
+ proofs and certified programs using Coq. With its large collection of
+ examples and exercies it is an invaluable tool for researchers,
+ students, and engineers interested in formal methods and the
+ development of zerofault software."
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 93]{Bro93}
 author = "Bronstein, Manuel (ed)",
 year = "1993",
 month = "July"
 isbn = "0897916042",
ISSAC'93: proceedings of the 1993 International Symposium on Symbolic
and Algebraic Computation, Kiev, Ukraine,
ACM Press New York, NY 10036, USA, ISBN
LCCN QA76.95 I59 1993 ACM order number 505930
 keywords = "axiomref",
+\bibitem[Boulme 00]{BHR00} Boulm\'e, S.; Hardin, T.; Rioboo, R.
+``Polymorphic Data Types, Objects, Modules and Functors,: is it too much?''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/BHR00.pdf
+ abstract = "
+ Abstraction is a powerful tool for developers and it is offered by
+ numerous features such as polymorphism, classes, modules, and
+ functors, $\ldots$ A working programmer may be confused by this
+ abundance. We develop a computer algebra library which is being
+ certificed. Reporting this experience made with a language (Ocaml)
+ offering all these features, we argue that the are all needed
+ together. We compare several ways of using classes to represent
+ algebraic concepts, trying to follow as close as possible mathematical
+ specification. Thenwe show how to combine classes and modules to
+ produce code having very strong typing properties. Currently, this
+ library is made of one hundred units of functional code and behaves
+ faster than analogous ones such as Axiom."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Brunelli 08]{Brun08}
 author = "Brunelli, J.C.",
 title = "Streams and Lazy Evaluation Applied to Integrable Models",
 year = "2008",
 url = "http://arxiv.org/PS_cache/nlin/pdf/0408/0408058v1.pdf",
 paper = "Brun08.pdf",
 keywords = "axiomref",
+\bibitem[Boulme 01]{BHHMR01}
+Boulm\'e, S.; Hardin, T.; Hirschkoff, D.; M\'enissierMorain, V.; Rioboo, R.
+``On the way to certify Computer Algebra Systems''
+Calculemus2001
+%\verbaxiomdeveloper.org/axiomwebsite/papers/BHHMR01.pdf
+ abstract = "
+ The FOC project aims at supporting, within a coherent software system,
+ the entire process of mathematical computation, starting with proved
+ theories, ending with certified implementations of algorithms. In this
+ paper, we explain our design requirements for the implementation,
+ using polynomials as a running example. Indeed, proving correctness of
+ implementations depends heavily on the way this design allows
+ mathematical properties to be truly handled at the programming level.
+
+ The FOC project, started at the fall of 1997, is aimed to build a
+ programming environment for the development of certified symbolic
+ computation. The working languages are Coq and Ocaml. In this paper,
+ we present first the motivations of the project. We then explain why
+ and how our concern for proving properties of programs has led us to
+ certain implementation choices in Ocaml. This way, the sources express
+ exactly the mathematical dependencies between different structures.
+ This may ease the achievement of proofs."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Computer algebra procedures to manipulate pseudodifferential
operators are implemented to perform calculations with integrable
models. We use lazy evaluation and streams to represent and operate
with pseudodifferential operators. No order of truncation is needed
since terms are produced on demand. We give a series of concrete
examples using the computer algebra language MAPLE.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Daly 10]{Daly10} Daly, Timothy
+``Intel Instruction Semantics Generator''
+\verbdaly.axiomdeveloper.org/TimothyDaly_files/publications/sei/intel/intel.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Daly10.pdf
+ abstract = "
+ Given an Intel x86 binary, extract the semantics of the instruction
+ stream as Conditional Concurrent Assignments (CCAs). These CCAs
+ represent the semantics of each individual instruction. They can be
+ composed to represent higher level semantics."
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 93]{BS93}
 author = "Bronstein, Manuel and Salvy, Bruno",
 title = "Full Partial Fraction Decomposition of Rational Functions",
 year = "1993",
 pages = "157160",
 isbn = "0897916042",
In Bronstein [Bro93] LCCN QA76.95 I59 1993
 keywords = "axiomref",
+\bibitem[Danielsson 06]{Dani06} Danielsson, Nils Anders; Hughes, John;
+Jansson, Patrik; Gibbons, Jeremy
+``Fast and Loose Reasoning is Morally Correct''
+ACM POPL'06 January 2005, Charleston, South Carolina, USA
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dani06.pdf
+ abstract = "
+ Functional programmers often reason about programs as if they were
+ written in a total language, expecting the results to carry over to
+ nontoal (partial) languages. We justify such reasoning.
+
+ Two languages are defined, one total and one partial, with identical
+ syntax. The semantics of the partial language includes partial and
+ infinite values, and all types are lifted, including the function
+ spaces. A partial equivalence relation (PER) is then defined, the
+ domain of which is the total subset of the partial language. For types
+ not containing function spaces the PER relates equal values, and
+ functions are related if they map related values to related values.
+
+ It is proved that if two closed terms have the same semantics in the
+ total language, then they have related semantics in the partial
+ language. It is also shown that the PER gives rise to a bicartesian
+ closed category which can be used to reason about values in the domain
+ of the relation."
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Bro92a,
 author = "Bronstein, Manuel",
 title = "Integration and Differential Equations in Computer Algebra",
 year = "1992",
 url = "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.576",
 paper = "Bro92a.pdf",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Davenport 12]{Davenp12} Davenport, James H.; Bradford, Russell;
+England, Matthew; Wilson, David
+``Program Verification in the presence of complex numbers, functions with
+branch cuts etc.''
+\verbarxiv.org/pdf/1212.5417.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Davenp12.pdf
+ abstract = "
+ In considering the reliability of numerical programs, it is normal to
+ ``limit our study to the semantics dealing with numerical precision''.
+ On the other hand, there is a great deal of work on the reliability of
+ programs that essentially ignores the numerics. The thesis of this
+ paper is that there is a class of problems that fall between these
+ two, which could be described as ``does the lowlevel arithmetic
+ implement the highlevel mathematics''. Many of these problems arise
+ because mathematics, particularly the mathematics of the complex
+ numbers, is more difficult than expected: for example the complex
+ function log is not continuous, writing down a program to compute an
+ inverse function is more complicated than just solving an equation,
+ and many algebraic simplification rules are not universally valid.
+
+ The good news is that these problems are {\sl theoretically} capable
+ of being solved, and are {\sl practically} close to being solved, but
+ not yet solved, in several realworld examples. However, there is
+ still a long way to go before implementations match the theoretical
+ possibilities."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Dolzmann 97]{Dolz97} Dolzmann, Andreas; Sturm, Thomas
+``Guarded Expressions in Practice''
+\verbredlog.dolzmann.de/papers/pdf/MIP9702.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dolz97.pdf
+ abstract = "
+ Computer algebra systems typically drop some degenerate cases when
+ evaluating expressions, e.g. $x/x$ becomes 1 dropping the case
+ $x=0$. We claim that it is feasible in practice to compute also the
+ degenerate cases yielding {\sl guarded expressions}. We work over real
+ closed fields but our ideas about handling guarded expressions can be
+ easily transferred to other situations. Using formulas as guards
+ provides a powerful tool for heuristically reducing the combinatorial
+ explosion of cases: equivalent, redundant, tautological, and
+ contradictive cases can be detected by simplification and quantifier
+ elimination. Our approach allows to simplify the expressions on the
+ basis of simplification knowledge on the logical side. The method
+ described in this paper is implemented in the REDUCE package GUARDIAN,
+ which is freely available on the WWW."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe in this paper how the problems of computing indefinite
integrals and solving linear ordinary differential equations in closed
form are now solved by computer algebra systems. After a brief review
of the mathematical history of those problems, we outline the two
major algorithms for them (respectively the Risch and Singer
algorithms) and the recent improvements on those algorithms which has
allowed them to be implemented.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Dos Reis 11]{DR11} Dos Reis, Gabriel; Matthews, David; Li, Yue
+``Retargeting OpenAxiom to Poly/ML: Towards an Integrated Proof Assistants
+and Computer Algebra System Framework''
+Calculemus (2011) Springer
+\verbparadise.caltech.edu/~yli/paper/oapolyml.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/DR11.pdf
+ abstract = "
+ This paper presents an ongoing effort to integrate the Axiom family of
+ computer algebra systems with Poly/MLbased proof assistants in the
+ same framework. A long term goal is to make a large set of efficient
+ implementations of algebraic algorithms available to popular proof
+ assistants, and also to bring the power of mechanized formal
+ verification to a family of strongly typed computer algebra systems at
+ a modest cost. Our approach is based on retargeting the code generator
+ of the OpenAxiom compiler to the Poly/ML abstract machine."
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Beneke 94]{BS94}
 author = "Beneke, T. and Schwippert, W.",
 title = "Doubletrack into the future: MathCAD will gain new users with Standard and Plus versions",
 year = "1994",
 month = "July",
 pages = "107110",
 keywords = "axiomref",
Elektronik, 43(15) CODEN EKRKAR ISSN 00135658
+\bibitem[Dunstan 00a]{Dun00a} Dunstan, Martin N.
+``Adding Larch/Aldor Specifications to Aldor''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dunxx.pdf
+ abstract = "
+ We describe a proposal to add Larchstyle annotations to the Aldor
+ programming language, based on our PhD research. The annotations
+ are intended to be machinecheckable and may be used for a variety
+ of purposes ranging from compiler optimizations to verification
+ condition (VC) generation. In this report we highlight the options
+ available and describe the changes which would need to be made to
+ the compiler to make use of this technology."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 97a]{Bro97a}
 author = "Bronstein, Manuel and Weil, JacquesArthur",
 title = "On Symmetric Powers of Differential Operators",
 year = "1997",
 pages = "156163",
 keywords = "axiomref",
 url = "http://wwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html"
 paper = "Bro97a.pdf",
 publisher = "ACM, NY",
ISSAC'97
+\bibitem[Dunstan 98]{Dun98} Dunstan, Martin; Kelsey, Tom; Linton, Steve;
+Martin, Ursula
+``Lightweight Formal Methods For Computer Algebra Systems''
+\verbwww.cs.standrews.ac.uk/~tom/pub/issac98.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dun98.pdf
+ abstract = "
+ Demonstrates the use of formal methods tools to provide a semantics
+ for the type hierarchy of the Axiom computer algebra system, and a
+ methodology for Aldor program analysis and verification. There are
+ examples of abstract specifications of Axiom primitives."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present alternative algorithms for computing symmetric powers of
linear ordinary differential operators. Our algorithms are applicable
to operators with coefficients in arbitrary integral domains and
become faster than the traditional methods for symmetric powers of
sufficiently large order, or over sufficiently complicated coefficient
domains. The basic ideas are also applicable to other computations
involving cyclic vector techniques, such as exterior powers of
differential or difference operators.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Dunstan 99a]{Dun99a} Dunstan, MN
+``Larch/Aldor  A Larch BISL for AXIOM and Aldor''
+PhD Thesis, 1999
+\verbwww.cs.standrews.uk/files/publications/Dun99.php
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dun99a.pdf
+ abstract = "
+ In this thesis we investigate the use of lightweight formal methods
+ and verification conditions (VCs) to help improve the reliability of
+ components constructed within a computer algebra system. We follow the
+ Larch approach to formal methods and have designed a new behavioural
+ interface specification language (BISL) for use with Aldor: the
+ compiled extension language of Axiom and a fullyfeatured programming
+ language in its own right. We describe our idea of lightweight formal
+ methods, present a design for a lightweight verification condition
+ generator and review our implementation of a prototype verification
+ condition generator for Larch/Aldor."
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Borwein 00]{Bor00}
 author = "Borwein, Jonathan",
 title = "Multimedia tools for communicating mathematics",
 year = "2000",
 pages = "58",
 isbn = "3540424504",
 publisher = "SpringerVerlag",
 keywords = "axiomref"
+\bibitem[Dunstan 00]{Dun00} Dunstan, Martin; Kelsey, Tom; Martin, Ursula;
+Linton, Steve
+``Formal Methods for Extensions to CAS''
+FME'99, Toulouse, France, Sept 2024, 1999, pp 17581777
+\verbtom.host.cs.standrews.ac.uk/pub/fm99.ps
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dun00.pdf
+ abstract = "
+ We demonstrate the use of formal methods tools to provide a semantics
+ for the type hierarchy of the AXIOM computer algebra system, and a
+ methodology for Aldor program analysis and verification. We give a
+ case study of abstract specifications of AXIOM primitives, and provide
+ an interface between these abstractions and Aldor code."
\end{chunk}
\begin{chunk}{axiom.bib}
@article{BT94,
 author = "Brown, R. and Tonks, A.",
 title = "Calculations with simplicial and cubical groups in AXIOM",
 journal = "Journal of Symbolic Computation",
 volume = "17",
 number = "2",
 pages = "159179",
 year = "1994",
 month = "February",
 misc = "CODEN JSYCEH ISSN 07477171",
 keywords = "axiomref"
+@misc{Hard13,
+ author = "Hardin, David S. and McClurg, Jedidiah R. and Davis, Jennifer A.",
+ title = "Creating Formally Verified Components for Layered Assurance with an LLVM to ACL2 Translator",
+ url = "http://www.jrmcclurg.com/papers/law_2013_paper.pdf",
+ paper = "Hard13.pdf",
+ abstract = "
+ This paper describes an effort to create a library of formally
+ verified software component models from code that have been compiled
+ using the LowLevel Virtual Machine (LLVM) intermediate form. The idea
+ is to build a translator from LLVM to the applicative subset of Common
+ Lisp accepted by the ACL2 theorem prover. They perform verification of
+ the component model using ACL2's automated reasoning capabilities."
}
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Brow95,
 author = "Brown, Ronald and Dreckmann, Winfried",
 title = "Domains of data and domains of terms in AXIOM",
 year = "1995",
 keywords = "axiomref",
 paper = "DB95.pdf"
+@misc{Hard14,
+ author = "Hardin, David S. and Davis, Jennifer A. and Greve, David A. and
+ McClurg, Jedidiah R.",
+ title = "Development of a Translator from LLVM to ACL2",
+ url = "http://arxiv.org/pdf/1406.1566",
+ paper = "Hard14.pdf",
+ abstract = "
+ In our current work a library of formally verified software components
+ is to be created, and assembled, using the LowLevel Virtual Machine
+ (LLVM) intermediate form, into subsystems whose toplevel assurance
+ relies on the assurance of the individual components. We have thus
+ undertaken a project to build a translator from LLVM to the
+ applicative subset of Common Lisp accepted by the ACL2 theorem
+ prover. Our translator produces executable ACL2 formal models,
+ allowing us to both prove theorems about the translated models as well
+ as validate those models by testing. The resulting models can be
+ translated and certified without user intervention, even for code with
+ loops, thanks to the use of the def::ung macro which allows us to
+ defer the question of termination. Initial measurements of concrete
+ execution for translated LLVM functions indicate that performance is
+ nearly 2.4 million LLVM instructions per second on a typical laptop
+ computer. In this paper we overview the translation process and
+ illustrate the translator's capabilities by way of a concrete example,
+ including both a functional correctness theorem as well as a
+ validation test for that example."
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The main new concept we wish to illustrate in this paper is a
distinction between ``domains of data'' and ``domains of terms'', and
its use in the programming of certain mathematical structures.
Although this distinction is implicit in much of the programming work
that has gone into the construction of Axiom categories and domains,
we believe that a formalisation of this is new, that standards and
conventions are necessary and will be useful in various other
contexts. We shall show how this concept may be used for the coding of
free categories and groupoids on directed graphs.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Lamport 02]{Lamp02} Lamport, Leslie
+``Specifying Systems''
+\verbresearch.microsoft.com/enus/um/people/lamport/tla/book020808.pdf
+AddisonWesley ISBN 032114306X
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Lamp02.pdf
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Buchberger 85]{BC85} Buchberger, Bruno and Caviness, Bob F. (eds)
EUROCAL '85: European Conference on Computer Algebra, Linz, Austria,
LLCN QA155.7.E4 E86
 isbn = "0387159835, 0387159843",
 year = "1985",
 month = "April",
 publisher = "SpringerVerlag, Berlin, Germany",
 keywords = "axiomref",
 misc = "Lecture Notes in Computer Science, Vol 204",
+\bibitem[Martin 97]{Mart97} Martin, U.; Shand, D.
+``Investigating some Embedded Verification Techniques for Computer
+ Algebra Systems''
+\verbwww.risc.jku.at/conferences/Theorema/papers/shand.ps.gz
+%\verbaxiomdeveloper.org/axiomwebsite/papers/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}
\begin{chunk}{axiom.bib}
@misc{Buh05,
 author = "Buhl, Soren L.",
 title = "Some Reflections on Integrating a Computer Algebra System in R",
 year = "2005",
 keywords = "axiomref"
+@book{Maso86,
+ author = "Mason, Ian A.",
+ title = "The Semantics of Destructive Lisp",
+ publisher = "Center for the Study of Language and Information",
+ year = "1986",
+ isbn = "0937073067",
+ abstract = "
+ Our basic premise is that the ability to construct and modify programs
+ will not improve without a new and comprehensive look at the entire
+ programming process. Past theoretical research, say, in the logic of
+ programs, has tended to focus on methods for reasoning about
+ individual programs; little has been done, it seems to us, to develop
+ a sound understanding of the process of programming  the process by
+ which programs evolve in concept and in practice. At present, we lack
+ the means to describe the techniques of program construction and
+ improvement in ways that properly link verification, documentation and
+ adaptability."
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Burge 91]{Burg91}
 author = "Burge, W.H.",
 title = "Scratchpad and the RogersRamanujan identities",
 year = "1991",
 pages = "189190",
 isbn = "0897914376",
 keywords = "axiomref",
In Watt [Wat91], LCCN QA76.95.I59
+\bibitem[Newcombe 13]{Newc13} Newcombe, Chris; Rath, Tim; Zhang, Fan;
+Munteanu, Bogdan; Brooker, Marc; Deardeuff, Michael
+``Use of Formal Methods at Amazon Web Services''
+\verbresearch.microsoft.com/enus/um/people/lamport/tla/
+\verbformalmethodsamazon.pdf
+ abstract = "
+ In order to find subtle bugs in a system design, it is necessary to
+ have a precise description of that design. There are at least two
+ major benefits to writing a precise design; the author is forced to
+ think more clearly, which helps eliminate ``plausible handwaving'',
+ and tools can be applied to check for errors in the design, even while
+ it is being written. In contrast, conventional design documents
+ consist of prose, static diagrams, and perhaps pseudocode in an ad
+ hoc untestable language. Such descriptions are far from precise; they
+ are often ambiguous, or omit critical aspects such as partial failure
+ or the granularity of concurrency (i.e. which constructs are assumed
+ to be atomic). At the other end of the spectrum, the final executable
+ code is unambiguous, but contains an overwhelming amount of detail. We
+ needed to be able to capture the essence of a design in a few hundred
+ lines of precise description. As our designs are unavoidably complex,
+ we need a highlyexpressive language, far above the level of code, but
+ with precise semantics. That expressivity must cover realworld
+ concurrency and faulttolerance. And, as we wish to build services
+ quickly, we wanted a language that is simple to learn and apply,
+ avoiding esoteric concepts. We also very much wanted an existing
+ ecosystem of tools. We found what we were looking for in TLA+, a
+ formal specification language."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This note sketches the part played by Scratchpad in obtaining new
proofs of Euler's theorem and the RogersRamanujan Identities.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@techreport{BW87,
 author = "Burge, W. and Watt, S.",
 title = "Infinite structures in SCRATCHPAD II",
 year = "1987",
 institution = "IBM Research",
 type = "Technical Report",
 number = "RC 12794",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Poll 99a]{P99a} Poll, Erik
+``The Type System of Axiom''
+\verbwww.cs.ru.nl/E.Poll/talks/axiom.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/P99a.pdf
+ abstract = "
+ This is a slide deck from a talk on the correspondence between
+ Axiom/Aldor types and Logic."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Burge 87a]{BWM87}
 author = "Burge, William H. and Watt, Stephen M. and Morrison, Scott C.",
 title = "Streams and Power Series",
 year = "1987",
 pages = "912",
 keywords = "axiomref",
in [Wit87], pp912
+\bibitem[Poll 99]{PT99} Poll, Erik; Thompson, Simon
+``The Type System of Aldor''
+\verbwww.cs.kent.ac.uk/pubs/1999/874/content.ps
+%\verbaxiomdeveloper.org/axiomwebsite/papers/PT99.pdf
+ abstract = "
+ This paper gives a formal description of  at least a part of 
+ the type system of Aldor, the extension language of the Axiom.
+ In the process of doing this a critique of the design of the system
+ emerges."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Burge 89]{BW89}
 author = "Burge, W. H. and Watt, S. M.",
 title = "Infinite structures in Scratchpad II",
 year = "1989",
 pages = "138148",
 isbn = "3540515178",
 keywords = "axiomref",
in Davenport [Dav89], LCCN QA155.7.E4E86 1987

\end{chunk}
+\bibitem[Poll (a)]{PTxx} Poll, Erik; Thompson, Simon
+``Adding the axioms to Axiom. Toward a system of automated reasoning in
+Aldor''
+\verbciteseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.1457&rep=rep1&type=ps
+%\verbaxiomdeveloper.org/axiomwebsite/papers/PTxx.pdf
+ abstract = "
+ This paper examines the proposal of using the type system of Axiom to
+ represent a logic, and thus to use the constructions of Axiom to
+ handle the logic and represent proofs and propositions, in the same
+ way as is done in theorem provers based on type theory such as Nuprl
+ or Coq.
\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ The paper shows an interesting way to decorate Axiom with pre and
+ postconditions.
\begin{chunk}{ignore}
\bibitem[Calmet 94]{Cal94} Calmet, J. (ed)
Rhine Workshop on Computer Algebra, Proceedings.
Universit{\"a}t Karsruhe, Karlsruhe, Germany 1994
 keywords = "axiomref",
+ The CurryHoward correspondence used is
+ \begin{verbatim}
+ PROGRAMMING LOGIC
+ Type Formula
+ Program Proof
+ Product/record type (...,...) Conjunction
+ Sum/union type \/ Disjunction
+ Function type > Implication
+ Dependent function type (x:A) > B(x) Universal quantifier
+ Dependent product type (x:A,B(x)) Existential quantifier
+ Empty type Exit Contradictory proposition
+ One element type Triv True proposition
+ \end{verbatim}"
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Camion 92]{CCM92}
 author = "Camion, Paul and Courteau, Bernard and Montpetit, Andre",
 title = "A combinatorial problem in Hamming Graphs and its solution in Scratchpad",
 year = "1992",
 month = "January",
 keywords = "axiomref",
Rapports de recherche 1586, Institut National de Recherche en
Informatique et en Automatique, Le Chesnay, France, 12pp
+\bibitem[Poll 00]{PT00} Poll, Erik; Thompson, Simon
+``Integrating Computer Algebra and Reasoning through the Type System
+of Aldor''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/PT00.pdf
+ abstract = "
+ A number of combinations of reasoning and computer algebra systems
+ have been proposed; in this paper we describe another, namely a way to
+ incorporate a logic in the computer algebra system Axiom. We examine
+ the type system of Aldor  the Axiom Library Compiler  and show
+ that with some modifications we can use the dependent types of the
+ system to model a logic, under the CurryHoweard isomorphism. We give
+ a number of example applications of the logi we construct and explain
+ a prototype implementation of a modified typechecking system written
+ in Haskell."
\end{chunk}
+\subsection{Interval Arithmetic} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Caprotti 00]{CCR00}
 author = "Caprotti, Olga and Cohen, Arjeh M. and Riem, Manfred",
 title = "Java Phrasebooks for Computer Algebra and Automated Deduction",
 url = "http://www.sigsam.org/bulletin/articles/132/paper8.pdf",
 paper = "CCR00.pdf",
 keywords = "axiomref",
+\bibitem[Boehm 86]{Boe86} Boehm, HansJ.; Cartwright, Robert; Riggle, Mark;
+O'Donnell, Michael J.
+``Exact Real Arithmetic: A Case Study in Higher Order Programming''
+\verbdev.acm.org/pubs/citations/proceedings/lfp/319838/p162boehm
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Boe86.pdf
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{CC99,
 author = "Capriotti, O. and Carlisle, D.",
 title = "OpenMath and MathML: Semantic Mark Up for Mathematics",
 year = "1999",
 url = "http://www.acm.org/crossroads/xrds62/openmath.html",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Briggs 04]{Bri04} Briggs, Keith
+``Exact real arithmetic''
+\verbkeithbriggs.info/documents/xrkenttalkpp.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bri04.pdf
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Capr99,
 author = "Capriotti, Olga and Cohen, Arjeh M. and Cuypers, Hans and Sterk, Hans",
 title = "OpenMath Technology for Interactive Mathematical Documents",
 year = "2002",
 pages = "5166",
 publisher = "SpringerVerlag, Berlin, Germany",
 url = "http://www.win.tue.nl/~hansc/lisbon.pdf",
 paper = "Capr99.pdf",
 misc = "in Multimedia Tools for Communicating Mathematics",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Fateman 94]{Fat94} Fateman, Richard J.; Yan, Tak W.
+``Computation with the Extended Rational Numbers and an Application to
+Interval Arithmetic''
+\verbwww.cs.berkeley.edu/~fateman/papers/extrat.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat94.pdf
+ abstract = "
+ Programming languages such as Common Lisp, and virtually every
+ computer algebra system (CAS), support exact arbitraryprecision
+ integer arithmetic as well as exect rational number computation.
+ Several CAS include interval arithmetic directly, but not in the
+ extended form indicated here. We explain why changes to the usual
+ rational number system to include infinity and ``notanumber'' may be
+ useful, especially to support robust interval computation. We describe
+ techniques for implementing these changes."
\end{chunk}
\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",
 keywords = "axiomref"
+@incollection{Lamb06,
+ author = "Lambov, Branimir",
+ title = "Interval Arithmetic Using SSE2",
+ booktitle = "Lecture Notes in Computer Science",
+ publisher = "SpringerVerlag",
+ year = "2006",
+ isbn = "9783540855200",
+ pages = "102113"
}
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Chu85,
 author = "Chudnovsky, D.V and Chudnovsky, G.V.",
 title = "Elliptic Curve Calculations in Scratchpad II",
 year = "1985",
 institution = "Mathematics Dept., IBM Research",
 type = "Scratchpad II Newsletter 1 (1)",
 keywords = "axiomref"
}
+\subsection{Numerics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Atkinson 09]{Atk09} Atkinson, Kendall; Han, Welmin; Stewear, David
+``Numerical Solution of Ordinary Differential Equations''
+\verbhomepage.math.uiowa.edu/~atkinson/papers/NAODE_Book.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Atk09.pdf
+ abstract = "
+ This book is an expanded version of supplementary notes that we used
+ for a course on ordinary differential equations for upperdivision
+ undergraduate students and beginning graduate students in mathematics,
+ engineering, and sciences. The book introduces the numerical analysis
+ of differential equations, describing the mathematical background for
+ understanding numerical methods and giving information on what to
+ expect when using them. As a reason for studying numerical methods as
+ a part of a more general course on differential equations, many of the
+ basic ideas of the numerical analysis of differential equations are
+ tied closely to theoretical behavior associated with the problem being
+ solved. For example, the criteria for the stability of a numerical
+ method is closely connected to the stability of the differential
+ equation problem being solved."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Chudnovsky 87]{Chu87}
 author = "Chudnovsky, D.V and Chudnovsky, G.V.",
 title = "New Analytic Methods of Polynomial Root Finding",
 year = "1987",
 pages = "2",
 keywords = "axiomref",
in [Wit87]
+\bibitem[Crank 96]{Cran96} Crank, J.; Nicolson, P.
+``A practical method for numerical evaluations of solutions of partial
+ differential equations of heatconduction type''
+Advances in Computational Mathematics Vol 6 pp207226 (1996)
+\verbwww.acms.arizona.edu/FemtoTheory/MK_personal/opti547/literature/
+\verbCNMethodoriginal.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Cran96.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Chudnovsky 89]{Chu89}
 author = "Chudnovsky, D.V. and Chudnovsky, G.V.",
 title = "The computation of classical constants",
 year = "1989",
 month = "November",
 pages = "81788182",
 keywords = "axiomref",
Proc. Natl. Acad. Sci. USA Vol 86
+\bibitem[Lef\'evre 06]{Lef06} Lef\'evre, Vincent; Stehl\'e, Damien;
+Zimmermann, Paul
+``Worst Cases for the Exponential Function
+in the IEEE754r decimal64 Format''
+in Lecture Notes in Computer Science, Springer ISBN 9783540855200
+(2006) pp114125
+ abstract = "
+ We searched for the worst cases for correct rounding of the
+ exponential function in the IEEE 754r decimal64 format, and computed
+ all the bad cases whose distance from a breakpoint (for all rounding
+ modes) is less than $10^{15}$ ulp, and we give the worst ones. In
+ particular, the worst case for
+ $\vert{}x\vert{} \ge 3 x 10^{11}$ is
+ \[
+ exp(9.407822313572878x10^{2} =
+ 1.09864568206633850000000000000000278\ldots
+ \]
+ This work can be extended to other elementary functions in the decimal64
+ format and allows the design of reasonably fast routines that will
+ evaluate these functions with correct rounding, at least in some
+ situations."
\end{chunk}
\begin{chunk}{axiom.bib}
@proceedings{CJ86,
 editor = "Chudnovsky, David and Jenks, Richard",
 title = "Computers in Mathematics",
 year = "1986",
 month = "July",
 isbn = "0824783417",
 note = "International Conference on Computers and Mathematics",
 publisher = "Marcel Dekker, Inc",
 keywords = "axiomref"
+@book{Hamm62,
+ author = "Hamming R W.",
+ title = "Numerical Methods for Scientists and Engineers",
+ publisher = "Dover",
+ year = "1973",
+ isbn = "0486652416"
}
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Cohe03,
 author = "Cohen, Arjeh and Cuypers, M. and Barreiro, Hans and Reinaldo, Ernesto and Sterk, Hans",
 title = "Interactive Mathematical Documents on the Web",
 year = "2003",
 pages = "289306",
 editor = "Joswig, M. and Takayma, N.",
 publisher = "SpringerVerlag, Berlin, Germany",
 keywords = "axiomref",
 misc = "in Algebra, Geometry and Software Systems"
}
+\subsection{Advanced Documentation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem [Bostock 14]{Bos14} Bostock, Mike
+``Visualizing Algorithms''
+\verbbost.ocks.org/mike/algorithms
+ abstract = "
+ This website hosts various ways of visualizing algorithms. The hope is
+ that these kind of techniques can be applied to Axiom."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cohen 91]{CC91} Cohen, G.; Charpin, P.; (ed)
EUROCODE '90 International Symposium on
Coding Theory and Applications Proceedings. SpringerVerlag, Berlin, Germany
/ Heidelberg, Germany / London, UK / etc., 1991 ISBN 0387543031
(New York), 3540543031 (Berlin), LCCN QA268.E95 1990
 keywords = "axiomref",
+\bibitem[Leeuwen]{Leexx} van Leeuwen, Andr\'e M.A.
+``Representation of mathematical object in interactive books''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Leexx.pdf
+ abstract = "
+ We present a model for the representation of mathematical objects in
+ structured electronic documents, in a way that allows for interaction
+ with applications such as computer algebra systems and proof checkers.
+ Using a representation that reflects only the intrinsic information of
+ an object, and storing applicationdependent information in socalled
+ {\sl application descriptions}, it is shown how the translation from
+ the internal to an external representation and {\sl vice versa} can be
+ achieved. Hereby a formalisation of the concept of {\sl context} is
+ introduced. The proposed scheme allows for a high degree of
+ application integration, e.g., parallel evaluation of subexpressions
+ (by different computer algebra systems), or a proof checker using a
+ computer algebra system to verify an equation involving a symbolic
+ computation."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Conrad (a)]{CFMPxxa}
 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",
+\bibitem[Soiffer 91]{Soif91} Soiffer, Neil Morrell
+``The Design of a User Interface for Computer Algebra Systems''
+\verbwww.eecs.berkeley.edu/Pubs/TechRpts/1991/CSD91626.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Soif91.pdf
+ abstract = "
+ This thesis discusses the design and implementation of natural user
+ interfaces for Computer Algebra Systems. Such an interface must not
+ only display expressions generated by the Computer Algebra System in
+ standard mathematical notation, but must also allow easy manipulation
+ and entry of expressions in that notation. The user interface should
+ also assist in understanding of large expressions that are generated
+ by Computer Algebra Systems and should be able to accommodate new
+ notational forms."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}

\begin{chunk}{axiom.bib}
@misc{CFMPxxb,
 author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra",
 title = "Mathematical Use Cases lead naturally to nonstandard Inheritance Relationships: How to make them accessible in a mainstream language?",
 paper = "CFMPxxb.pdf",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Victor 11]{Vict11} Victor, Bret
+``Up and Down the Ladder of Abstraction''
+\verbworrydream.com/LadderOfAbstraction
+ abstract = "
+ This interactive essay presents the ladder of abstraction, a technique for
+ thinking explicitly about these levels, so a designer can move among
+ them consciously and confidently. "
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Conceptually there is a strong correspondence between Mathematical
Reasoning and ObjectOriented techniques. We investigate how the ideas
of Method Renaming, Dynamic Inheritance and Interclassing can be used
to strengthen this relationship. A discussion is initiated concerning
the feasibility of each of these features.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@misc{Cuyp10,
 author = "Cuypers, Hans and Hendriks, Maxim and Knopper, Jan Willem",
 title = "Interactive Geometry inside MathDox",
 year = "2010",
 url = "http://www.win.tue.nl/~hansc/MathDox_and_InterGeo_paper.pdf",
 paper = "Cuyp10",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Victor 12]{Vict12} Victor, Bret
+``Inventing on Principle''
+\verbwww.youtube.com/watch?v=PUv66718DII
+ abstract = "
+ This video raises the level of discussion about humancomputer
+ interaction from a technical question to a question of effectively
+ capturing ideas. In particular, this applies well to Axiom's focus on
+ literate programming."
\end{chunk}
\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Differential Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@inproceedings{Dalm97,
 author = {Dalmas, St\'ephane and Ga\"etano, Marc and Watt, Stephen},
 title = "An OpenMath 1.0 Implementation",
 booktitle = "Proc. 1997 Int. Symp. on Symbolic and Algebraic Computation",
 series = "ISSAC'97",
 year = "1997",
 isbn = "0897918754",
 location = "Kihei, Maui, Hawaii, USA",
 pages = "241248",
 numpages = "8",
 url = "http://doi.acm.org/10.1145/258726.258794",
 doi = "10.1145/258726.258794",
 acmid = "258794",
 publisher = "ACM, New York, NY USA",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Abramov 95]{Abra95} Abramov, Sergei A.; Bronstein, Manuel;
+Petkovsek, Marko
+``On Polynomial Solutions of Linear Operator Equations''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Abra95.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dalmas 92]{Dal92} Dalmas, S.
``A polymorphic functional language applied to symbolic computation''
In Wang [Wan92] pp369375 ISBN 0897914899 (soft cover) 0897914902
(hard cover) LCCN QA76.95.I59 1992
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{axiom.bib}
@misc{Daly88,
 author = "Daly, Timothy",
 title = "Axiom in an Educational Setting, Axiom course slide deck",
 year = "1988",
 month = "January",
 keywords = "axiomref"
}
+\bibitem[Abramov 01]{Abra01} Abramov, Sergei; Bronstein, Manuel
+``On Solutions of Linear Functional Systems''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Abra01.pdf
+ abstract = "
+ We describe a new direct algorithm for transforming a linear system of
+ recurrences into an equivalent one with nonsingular leading or
+ trailing matrix. Our algorithm, which is an improvement to the EG
+ elimination method, uses only elementary linear algebra operations
+ (ranks, kernels, and determinants) to produce an equation satisfied by
+ the degress of the solutions with finite support. As a consequence, we
+ can boudn and compute the polynomial and rational solutions of very
+ general linear functional systems such as systems of differential or
+ ($q$)difference equations."
\end{chunk}
\begin{chunk}{ignore}TPDHERE
\bibitem[Daly 02]{Dal02} Daly, Timothy
``Axiom as open source''
SIGSAM Bulletin (ACM Special Interest Group
on Symbolic and Algebraic Manipulation) 36(1) pp28?? March 2002
CODEN SIGSBZ ISSN 01635824
 keywords = "axiomref",
+\begin{chunk}{ignore}
+\bibitem[Bronstein 96b]{Bro96b} Bronstein, Manuel
+``On the Factorization of Linear Ordinary Differential Operators''
+Mathematics and Computers in Simulation 42 pp 387389 (1996)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro96b.pdf
+ abstract = "
+ After reviewing the arithmetic of linear ordinary differential
+ operators, we describe the current status of the factorisation
+ algorithm, specially with respect to factoring over nonalgebraically
+ closed constant fields. We also describe recent results from Singer
+ and Ulmer that reduce determining the differential Galois group of an
+ operator to factoring."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Daly 03]{Dal03} Daly, Timothy
``The Axiom Wiki Website''
\verbaxiom.axiomdeveloper.org
 keywords = "axiomref",
+\bibitem[Bronstein 96a]{Bro96a} Bronstein, Manuel; Petkovsek, Marko
+``An introduction to pseudolinear algebra''
+Theoretical Computer Science V157 pp333 (1966)
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro96a.pdf
+ abstract = "
+ Pseudolinear algebra is the study of common properties of linear
+ differential and difference operators. We introduce in this paper its
+ basic objects (pseudoderivations, skew polynomials, and pseudolinear
+ operators) and describe several recent algorithms on them, which, when
+ applied in the differential and difference cases, yield algorithms for
+ uncoupling and solving systems of linear differential and difference
+ equations in closed form."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Daly 06]{Dal06} Daly, Timothy
``Axiom Volume 1: Tutorial''
Lulu, Inc. 860 Aviation Parkway,
Suite 300, Morrisville, NC 27560 USA, 2006 ISBN 141166597X 287pp
\verbwww.lulu.com/content/190827
 keywords = "axiomref",
+\bibitem[Bronstein xb]{Broxb} Bronstein, Manuel
+``Computer Algebra Algorithms for Linear Ordinary Differential and
+Difference equations''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/ecm3.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Broxb.pdf
+ abstract = "
+ Galois theory has now produced algorithms for solving linear ordinary
+ differential and difference equations in closed form. In addition,
+ recent algorithmic advances have made those algorithms effective and
+ implementable in computer algebra systems. After introducing the
+ relevant parts of the theory, we describe the latest algorithms for
+ solving such equations."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Daly 09]{Dal09} Daly, Timothy
``The Axiom Literate Documentation''
\verbaxiomdeveloper.org/axiomwebsite/documentation.html
 keywords = "axiomref",
+\bibitem[Bronstein 94]{Bro94} Bronstein, Manuel
+``An improved algorithm for factoring linear ordinary differential
+operators''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+ abstract = "
+ We describe an efficient algorithm for computing the associated
+ equations appearing in the BekeSchlesinger factorisation method for
+ linear ordinary differential operators. This algorithm, which is based
+ on elementary operations with sets of integers, can be easily
+ implemented for operators of any order, produces several possible
+ associated equations, of which only the simplest can be selected for
+ solving, and often avoids the degenerate case, where the order of the
+ associated equation is less than in the generic case. We conclude with
+ some fast heuristics that can produce some factorizations while using
+ only linear computations."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Daly 13]{Dal13} Daly, Timothy
``Literate Programming in the Large''
April 89, 2013 Portland Oregon
\verbconf.writethedocs.org
\verbdaly.axiomdeveloper.org
\verbwww.youtube.com/watch?v=Av0PQDVTP4A
 keywords = "axiomref",
+\bibitem[Bronstein 90]{Bro90} Bronstein, Manuel
+``On Solutions of Linear Ordinary Differential Equations in their
+Coefficient Field''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro90.pdf
+ abstract = "
+ We describe a rational algorithm for finding the denominator of any
+ solution of a linear ordinary differential equation in its coefficient
+ field. As a consequence, there is now a rational algorithm for finding
+ all such solutions when the coefficients can be built up from the
+ rational functions by finitely many algebraic and primitive
+ adjunctions. This also eliminates one of the computational bottlenecks
+ in algorithms that either factor or search for Liouvillian solutions
+ of such equations with Liouvillian coefficients."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 79a]{Dav79a} Davenport, J.H.
``What can SCRATCHPAD/370 do?''
VM/370 SPAD.SCRIPTS August 24, 1979 SPAD.SCRIPT
 keywords = "axiomref",
+\bibitem[Bronstein 96]{Bro96} Bronstein, Manuel
+``$\sum^{IT}$  A stronglytyped embeddable computer algebra library''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro96.pdf
+ abstract = "
+ We describe the new computer algebra library $\sum^{IT}$ and its
+ underlying design. The development of $\sum^{IT}$ is motivated by the
+ need to provide highly efficient implementations of key algorithms for
+ linear ordinary differential and ($q$)difference equations to
+ scientific programmers and to computer algebra users, regardless of
+ the programming language or interactive system they use. As such,
+ $\sum^{IT}$ is not a computer algebra system per se, but a library (or
+ substrate) which is designed to be ``plugged'' with minimal efforts
+ into different types of client applications."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 80]{Dav80} Davenport, J.H.; Jenks, R.D.
``MODLISP  an Introduction''
Proc LISP80, 1980, and IBM RC8357 Oct 1980
 keywords = "axiomref",
+\bibitem[Bronstein 99a]{Bro99a} Bronstein, Manuel
+``Solving linear ordinary differential equations over
+$C(x,e^{\int{f(x)dx}})$
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro99a.pdf
+ abstract = "
+ We describe a new algorithm for computing the solutions in
+ \[F=C(x,e^{\int{f(x)dx}})\] of linear ordinary differential equations
+ with coefficients in $F$. Compared to the general algorithm, our
+ algorithm avoids the computation of exponential solutions of equations
+ with coefficients in $C(x)$, as well as the solving of linear
+ differential systems over $C(x)$. Our method is effective and has been
+ implemented."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 84]{DGJ84} Davenport, J.; Gianni, P.; Jenks, R.;
Miller, V.; Morrison, S.; Rothstein, M.; Sundaresan, C.; Sutor, R.;
Trager, B.
``Scratchpad''
Mathematical Sciences Department, IBM Thomas Watson Research Center 1984
 keywords = "axiomref",
+\bibitem[Bronstein 00]{Bro00} Bronstein, Manuel
+``On Solutions of Linear Ordinary Differential Equations in their
+ Coefficient Field''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro00.pdf
+ abstract = "
+ We extend the notion of monomial extensions of differential fields,
+ i.e. simple transcendental extensions in which the polynomials are
+ closed under differentiation, to difference fields. The structure of
+ such extensions provides an algebraic framework for solving
+ generalized linear difference equations with coefficients in such
+ fields. We then describe algorithms for finding the denominator of any
+ solution of those equations in an important subclass of monomial
+ extensions that includes transcendental indefinite sums and
+ products. This reduces the general problem of finding the solutions of
+ such equations in their coefficient fields to bounding their
+ degrees. In the base case, this yields in particular a new algorithm
+ for computing the rational solutions of $q$difference equations with
+ polynomial coefficients."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 84a]{Dav84a} Davenport, James H.
``A New Algebra System''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav84a.pdf
 keywords = "axiomref",
+\bibitem[Bronstein 02]{Bro02} Bronstein, Manuel; Lafaille, S\'ebastien
+``Solutions of linear ordinary differential equations in terms of
+special functions''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro02.pdf
+ abstract = "
+ We describe a new algorithm for computing special function solutions
+ of the form $y(x) = m(x)F(\eta(x))$ of second order linear ordinary
+ differential equations, where $m(x)$ is an arbitrary Liouvillian
+ function, $\eta(x)$ is an arbitrary rational function, and $F$
+ satisfies a given second order linear ordinary differential
+ equations. Our algorithm, which is base on finding an appropriate
+ point transformation between the equation defining $F$ and the one to
+ solve, is able to find all rational transformations for a large class
+ of functions $F$, in particular (but not only) the $_0F_1$ and $_1F_1$
+ special functions of mathematical physics, such as Airy, Bessel,
+ Kummer and Whittaker functions. It is also able to identify the values
+ of the parameters entering those special functions, and can be
+ generalized to equations of higher order."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 85]{Dav85} Davenport, James H.
``The LISP/VM Foundation of Scratchpad II''
The Scratchpad II Newsletter, Volume 1, Number 1, September 1, 1985
IBM Corporation, Yorktown Heights, NY
 keywords = "axiomref",
+\bibitem[Bronstein 03]{Bro03} Bronstein, Manuel; Trager, Barry M.
+``A Reduction for Regular Differential Systems''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mega2003.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro03.pdf
+ abstract = "
+ We propose a definition of regularity of a linear differential system
+ with coefficients in a monomial extension of a differential field, as
+ well as a global and truly rational (i.e. factorisationfree)
+ iteration that transforms a system with regular finite singularites
+ into an equivalent one with simple finite poles. We then apply our
+ iteration to systems satisfied by bases of algebraic function fields,
+ obtaining algorithms for computing the number of irreducible
+ components and the genus of algebraic curves."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 88]{DST88} Davenport, J.H.; Siret, Y.; Tournier, E.
Computer Algebra: Systems and Algorithms for Algebraic Computation.
Academic Press, New York, NY, USA, 1988, ISBN 0122042329
\verbstaff.bath.ac.uk/masjhd/masternew.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/DST88.pdf
 keywords = "axiomref",
+\bibitem[Bronstein 03a]{Bro03a} Bronstein, Manuel; Sol\'e, Patrick
+``Linear recurrences with polynomial coefficients''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro03a.pdf
+ abstract = "
+ We relate sequences generated by recurrences with polynomial
+ coefficients to interleaving and multiplexing of sequences generated
+ by recurrences with constant coefficients. In the special case of
+ finite fields, we show that such sequences are periodic and provide
+ linear complexity estimates for all three constructions."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 14]{Dav14} Davenport, James H.
``Computer Algebra textbook''
\verbstaff.bath.ac.uk/masjhd/JHDCA.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav14.pdf
 keywords = "axiomref",
+\bibitem[Bronstein 05]{Bro05} Bronstein, Manuel; Li, Ziming; Wu, Min
+``PicardVessiot Extensions for Linear Functional Systems''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/issac2005.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro05.pdf
+ abstract = "
+ PicardVessiot extensions for ordinary differential and difference
+ equations are well known and are at the core of the associated Galois
+ theories. In this paper, we construct fundamental matrices and
+ PicardVessiot extensions for systems of linear partial functional
+ equations having finite linear dimension. We then use those extensions
+ to show that all the solutions of a factor of such a system can be
+ completed to solutions of the original system."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 89]{Dav89} Davenport, J.H. (ed)
EUROCAL '87 European Conference on Computer Algebra Proceedings
SpringerVerlag, Berlin, Germany / Heidelberg, Germany / London,
UK / etc., 1989 ISBN 3540515178 LCCN QA155.7.E4E86 1987
 keywords = "axiomref",
+\bibitem[Davenport 86]{Dav86} Davenport, J.H.
+``The Risch Differential Equation Problem''
+SIAM J. COMPUT. Vol 15, No. 4 1986
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav86.pdf
+ abstract = "
+ We propose a new algorithm, similar to Hermite's method for the
+ integration of rational functions, for the resolution of Risch
+ differential equations in closed form, or proving that they have no
+ resolution. By requiring more of the presentation of our differential
+ fields (in particular that the exponentials be weakly normalized), we
+ can avoid the introduction of arbitrary constants which have to be
+ solved for later.
+
+ We also define a class of fields known as exponentially reduced, and
+ show that solutions of Risch differential equations which arise from
+ integrating in these fields satisfy the ``natural'' degree constraints
+ in their main variables, and we conjecture (after Risch and Norman)
+ that this is true in all variables."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 90]{DT90} Davenport, J. H.; Trager, B. M.
``Scratchpad's view of algebra I: Basic commutative algebra''
In Miola [Mio90], pp4054. ISBN 0387525319 (New York),
3540525319 (Berlin). LCCN QA76.9.S88I576 1990 also in AXIOM Technical
Report, ATR/1, NAG Ltd., Oxford, 1992
 keywords = "axiomref",
+\bibitem[Singer 9]{Sing91.pdf} singer, Michael F.
+``Liouvillian Solutions of Linear Differential Equations with Liouvillian
+ Coefficients''
+J. Symbolic Computation V11 No 3 pp251273 (1991)
+\verbwww.sciencedirect.com/science/article/pii/S074771710880048X
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Sing91.pdf
+ abstract = "
+ Let $L(y)=b$ be a linear differential equation with coefficients in a
+ differential field $K$. We discuss the problem of deciding if such an
+ equation has a nonzero solution in $K$ and give a decision procedure
+ in case $K$ is an elementary extension of the field of rational
+ functions or is an algebraic extension of a transcendental liouvillian
+ extension of the field of rational functions We show how one can use
+ this result to give a procedure to find a basis for the space of
+ solutions, liouvillian over $K$, of $L(y)=0$ where $K$ is such a field
+ and $L(y)$ has coefficients in $K$."
\end{chunk}
\begin{chunk}{axiom.bib}
@inproceedings{Dave91,
 author = "Davenport, J. H. and Gianni, P. and Trager, B. M.",
 title = "Scratchpad's View of Algebra II: A Categorical View of Factorization",
 booktitle = "Proc. 1991 International Symposium on Symbolic and Algebraic Computation",
 series = "ISSAC '91",
 year = "1991",
 isbn = "0897914376",
 location = "Bonn, West Germany",
 pages = "3238",
 numpages = "7",
 url = "http://doi.acm.org/10.1145/120694.120699",
 doi = "10.1145/120694.120699",
 acmid = "120699",
 publisher = "ACM",
 address = "New York, NY, USA",
 keywords = "axiomref",
 paper = "Dave91.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Von Mohrenschildt 94]{Mohr94} Von Mohrenschildt, Martin
+``Symbolic Solutions of Discontinuous Differential Equations''
+\verbecollection.library.ethz.ch/eserv/eth:39463/eth3946301.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Mohr94.pdf
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper explains how Scratchpad solves the problem of presenting a
categorical view of factorization in unique factorization domains, i.e.
a view which can be propagated by functors such as SparseUnivariatePolynomial
or Fraction. This is not easy, as the constructive version of the classical
concept of UniqueFactorizationDomain cannot be so propagated. The solution
adopted is based largely on Seidenberg's conditions (F) and (P), but there
are several additional points that have to be borne in mind to produce
reasonably efficient algorithms in the required generality.

The consequence of the algorithms and interfaces presented in this
paper is that Scratchpad can factorize in any extension of the
integers or finite fields by any combination of polynomial, fraction
and algebraic extensions: a capability far more general than any other
computer algebra system possesses. The solution is not perfect: for
example we cannot use these general constructions to factorize
polyinmoals in $\overline{Z[\sqrt{5}]}[x]$ since the domain
$Z[\sqrt{5}]$ is not a unique factorization domain, even though
$\overline{Z[\sqrt{5}]}$ is, since it is a field. Of course, we can
factor polynomials in $\overline{Z}[\sqrt{5}][x]$
\end{adjustwidth}


\begin{chunk}{ignore}
\bibitem[Davenport 92]{DGT92} Davenport, J. H.;, Gianni, P.; Trager, B. M.
``Scratchpad's view of algebra II: A categorical view of factorization''
Technical Report TR4/92 (ATR/2) (NP2491), Numerical Algorithms Group, Inc.,
Downer's Grove, IL, USA and Oxford, UK, December 1992
\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
 keywords = "axiomref",
+\bibitem[Von Mohrenschildt 98]{Mohr98} Von Mohrenschildt, Martin
+``A Normal Form for Function Rings of Piecewise Functions''
+J. Symbolic Computation (1998) Vol 26 pp607619
+\verbwww.cas.mcmaster.ca/~mohrens/JSC.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Mohr98.pdf
+ abstract = "
+ Computer algebra systems often have to deal with piecewise continuous
+ functions. These are, for example, the absolute value function,
+ signum, piecewise defined functions but also functions that are the
+ supremum or infimum of two functions. We present a new algebraic
+ approach to these types of problems. This paper presents a normal form
+ for a function ring containing piecewise polynomial functions of an
+ expression. The main result is that this normal form can be used to
+ decide extensional equality of two piecewise functions. Also we define
+ supremum and infimum for piecewise functions; in fact, we show that
+ the function ring forms a lattice. Additionally, a method to solve
+ equalities and inequalities in this function ring is
+ presented. Finally, we give a ``user interface'' to the algebraic
+ representation of the piecewise functions."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 92a]{Dav92a} Davenport, J. H.
``The AXIOM system''
AXIOM Technical Report TR5/92 (ATR/3)
(NP2492) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and
Oxford, UK, December 1992
\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
 keywords = "axiomref",
+\bibitem[Weber 06]{Webe06} Weber, Andreas
+``Quantifier Elimination on Real Closed Fields and Differential Equations''
+\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber2006a.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe06.pdf
+ keywords = "survey",
+ abstract = "
+ This paper surveys some recent applications of quantifier elimination
+ on real closed fields in the context of differential
+ equations. Although polynomial vector fields give rise to solutions
+ involving the exponential and other transcendental functions in
+ general, many questions can be settled within the real closed field
+ without referring to the real exponential field. The technique of
+ quantifier elimination on real closed fields is not only of
+ theoretical interest, but due to recent advances on the algorithmic
+ side including algorithms for the simplification of quantifierfree
+ formulae the method has gained practical applications, e.g. in the
+ context of computing threshold conditions in epidemic modeling."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 92b]{Dav92b} Davenport, J. H.
``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
\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav92b.pdf
 keywords = "axiomref",
+\bibitem[Ulmer 03]{Ulm03} Ulmer, Felix
+``Liouvillian solutions of third order differential equations''
+J. Symbolic COmputations 36 pp 855889 (2003)
+\verbwww.sciencedirect.com/science/article/pii/S0747717103000658
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Ulm03.pdf
+ abstract = "
+ The Kovacic algorithm and its improvements give explicit formulae for
+ the Liouvillian solutions of second order linear differential
+ equations. Algorithms for third order differential equations also
+ exist, but the tools they use are more sophisticated and the
+ computations more involved. In this paper we refine parts of the
+ algorithm to find Liouvillian solutions of third order equations. We
+ show that,except for four finite groups and a reduction to the second
+ order case, it is possible to give a formula in the imprimitve
+ case. We also give necessary conditions and several simplifications
+ for the computation of the minimal polynomial for the remaining finite
+ set of finite groups (or any known finite group) by extracting
+ ramification information from the character table. Several examples
+ have been constructed, illustrating the possibilities and limitations."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}
+\subsection{Expression Simplification} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Davenport 92c]{DT92} Davenport, J. H.; Trager, B. M.
``Scratchpad's view of algebra I: Basic commutative algebra''
DISCO 90 Capri, Italy April 1990 ISBN 0387525319 pp4054
Technical Report TR3/92 (ATR/1)(NP2490), Numerical
Algorithms Group, Inc., Downer's Grove, IL, USA and Oxford, UK,
December 1992.
\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
 keywords = "axiomref",
+\bibitem[Carette 04]{Car04} Carette, Jacques
+``Understanding Expression Simplification''
+\verbwww.cas.mcmaster.ca/~carette/publications/simplification.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Car04.pdf
+ abstract = "
+ We give the first formal definition of the concept of {\sl
+ simplification} for general expressions in the context of Computer
+ Algebra Systems. The main mathematical tool is an adaptation of the
+ theory of Minimum Description Length, which is closely related to
+ various theories of complexity, such as Kolmogorov Complexity and
+ Algorithmic Information Theory. In particular, we show how this theory
+ can justify the use of various ``magic constants'' for deciding
+ between some equivalent representations of an expression, as found in
+ implementations of simplification routines."
\end{chunk}
+\subsection{Integration} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Davenport 93]{Dav93} Davenport, J. H.
``Primality testing revisited''
Technical Report TR2/93 (ATR/6)(NP2556) Numerical Algorithms Group, Inc.,
Downer's Grove, IL, USA and Oxford, UK, August 1993
\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
 keywords = "axiomref",
+\bibitem[Adamchik xx]{Adamxx} Adamchik, Victor
+``Definite Integration''
+\verbwww.cs.cmu.edu/~adamchik/articles/integr/mj.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Adamxx.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport (a)]{DFxx} Davenport, James; Faure, Christ\'ele
``The Unknown in Computer Algebra''
\verbaxiomwiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/DFxx.pdf
 keywords = "axiomref",
+\bibitem[Adamchik 97]{Adam97} Adamchik, Victor
+``A Class of Logarithmic Integrals''
+\verbwww.cs.cmu.edu/~adamchik/articles/issac/issac97.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Adam97.pdf
+ abstract = "
+ A class of definite integrals involving cyclotomic polynomials and
+ nested logarithms is considered. The results are given in terms of
+ derivatives of the Hurwitz Zeta function. Some special cases for which
+ such derivatives can be expressed in closed form are also considered."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Davenport 00]{Dav00} Davenport, James
``13th OpenMath Meeting''
James H. Davenport
``A New Algebra System''
May 1984
\verbxml.coverpages.org/openmath13.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav00.pdf
 keywords = "axiomref",
+\bibitem[Avgoustis 77]{Avgo77} Avgoustis, Ioannis Dimitrios
+``Definite Integration using the Generalized Hypergeometric Functions''
+\verbdspace.mit.edu/handle/1721.1/16269
+%\verbaxiomdeveloper.org/axiomwebsitep/papers/Avgo77.pdf
+ abstract = "
+ A design for the definite integration of approximately fifty Special
+ Functions is described. The Generalized Hypergeometric Functions are
+ utilized as a basis for the representation of the members of the above
+ set of Special Functions. Only a relatively small number of formulas
+ that generally involve Generalized Hypergeometric Functions are
+ utilized for the integration stage. A last and crucial stage is
+ required for the integration process: the reduction of the Generalized
+ Hypergeometric Function to Elementary and/or Special Functions.
+
+ The result of an early implementation which involves Laplace
+ transforms are given and some actual examples with their corresponding
+ timing are provided."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 12]{Dav12} Davenport, J.H.
``Computer Algebra''
\verbstaff.bath.ac.uk/masjhd/JHDCA.pdf
 keywords = "axiomref",
+\bibitem[Baddoura 89]{Bad89} Baddoura, Jamil
+``A Dilogarithmic Extension of Liouville's Theorem on Integration in Finite
+ Terms''
+\verbwww.dtic.mil/dtic/tr/fulltext/u2/a206681.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bad89.pdf
+ abstract = "
+ The result obtained generalizes Liouville's Theorem by allowing, in
+ addition to the elementary functions, dilogarithms to appear in the
+ integral of an elementary function. The basic conclusion is that an
+ associated function to the dilogarihm, if dilogarithms appear in the
+ integral, appears linearly, with logarithms appearing in a nonlinear
+ way."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport (b)]{DSTxx} Davenport, J. H.; Siret; Tournier
``Computer Algebra'' \hfill
\verbstaff.bath.ac.uk/masjhd/masternew.pdf
 keywords = "axiomref",
+\bibitem[Baddoura 94]{Bad94} Baddoura, Mohamed Jamil
+``Integration in Finite Terms with Elementary Functions and Dilogarithms''
+\verbdspace.mit.edu/bitstream/handle/1721.1/26864/30757785.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bad94.pdf
+ abstract = "
+ In this thesis, we report on a new theorem that generalizes
+ Liouville's theorem on integration in finite terms. The new theorem
+ allows dilogarithms to occur in the integral in addition to elementary
+ functions. The proof is base on two identities for the dilogarithm,
+ that characterize all the possible algebraic relations among
+ dilogarithms of functions that are built up from the rational
+ functions by taking transcendental exponentials, dilogarithms, and
+ logarithms."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dewar 94]{Dew94} Dewar, M. C.
``Manipulating Fortran Code in AXIOM and the AXIOMNAG Link''
Proceedings of the Workshop on Symbolic and Numeric Computing, ed by Apiola, H.
and Laine, M. and Valkeila, E. pp112 University of Helsinki, Finland (1994)
 keywords = "axiomref",
+\bibitem[Baddoura 10]{Bad10} Baddoura, Jamil
+``A Note on Symbolic Integration with Polylogarithms''
+J. Math Vol 8 pp229241 (2011)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bad10.pdf
+ abstract = "
+ We generalize partially Liouville's theorem on integration in finite
+ terms to allow polylogarithms of any order to occur in the integral in
+ addition to elementary functions. The result is a partial
+ generalization of a theorem proved by the author for the
+ dilogarithm. It is also a partial proof of a conjecture postulated by
+ the author in 1994. The basic conclusion is that an associated
+ function to the nth polylogarithm appears linearly with logarithms
+ appearing possibly in a polynomial way with nonconstant coefficients."
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Dewa,
 author = "Dewar, Mike",
 title = "OpenMath: An Overview",
 url = "http://www.sigsam.org/bulletin/articles/132/paper1.pdf",
 paper = "Dewa.pdf",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Bajpai 70]{Bajp70} Bajpai, S.D.
+``A contour integral involving legendre polynomial and Meijer's Gfunction''
+\verblink.springer.com/article/10.1007/BF03049565
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bajp70.pdf
+ abstract = "
+ In this paper a countour integral involving Legendre polynomial and
+ Meijer's Gfunction is evaluated. the integral is of general character
+ and it is a generalization of results recently given by Meijer,
+ MacRobert and others. An integral involving regular radial Coulomb
+ wave function is also obtained as a particular case."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dicrescenzo 89]{DD89} Dicrescenzo, C.; Duval, D.
``Algebraic extensions and algebraic closure in Scratchpad II''
In Gianni [Gia89], pp440446 ISBN 3540510842
LCCN QA76.95.I57 1998 Conference held jointly with AAECC6
 keywords = "axiomref",
+\bibitem[Bronstein 89]{Bro89a} Bronstein, M.
+``An Algorithm for the Integration of Elementary Functions''
+Lecture Notes in Computer Science Vol 378 pp491497 (1989)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro89a.pdf
+ abstract = "
+ Trager (1984) recently gave a new algorithm for the indefinite
+ integration of algebraic functions. His approach was ``rational'' in
+ the sense that the only algebraic extension computed in the smallest
+ one necessary to express the answer. We outline a generalization of
+ this approach that allows us to integrate mixed elementary
+ functions. Using only rational techniques, we are able to normalize
+ the integrand, and to check a necessary condition for elementary
+ integrability."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dingle 94]{Din94} Dingle, Adam; Fateman, Richard
``Branch Cuts in Computer Algebra''
1994 ISSAC, Oxford (UK), July 1994
\verbwww.cs.berkeley.edu/~fateman/papers/ding.ps
%\verbaxiomdeveloper.org/axiomwebsite/papers/Din94.pdf
 keywords = "axiomref",
+\bibitem[Bronstein 90a]{Bro90a} Bronstein, Manuel
+``Integration of Elementary Functions''
+J. Symbolic Computation (1990) 9, pp117173 September 1988
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro90a.pdf
+ abstract = "
+ We extend a recent algorithm of Trager to a decision procedure for the
+ indefinite integration of elementary functions. We can express the
+ integral as an elementary function or prove that it is not
+ elementary. We show that if the problem of integration in finite terms
+ is solvable on a given elementary function field $k$, then it is
+ solvable in any algebraic extension of $k(\theta)$, where $\theta$ is
+ a logarithm or exponential of an element of $k$. Our proof considers
+ an element of such an extension field to be an algebraic function of
+ one variable over $k$.
+
+ In his algorithm for the integration of algebraic functions, Trager
+ describes a Hermitetype reduction to reduce the problem to an
+ integrand with only simple finite poles on the associated Riemann
+ surface. We generalize that technique to curves over liouvillian
+ ground fields, and use it to simplify our integrands. Once the
+ multipe finite poles have been removed, we use the Puiseux expansions
+ of the integrand at infinity and a generalization of the residues to
+ compute the integral. We also generalize a result of Rothstein that
+ gives us a necessary condition for elementary integrability, and
+ provide examples of its use."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Many standard functions, such as the logarithms and square root
functions, cannot be defined continuously on the complex
plane. Mistaken assumptions about the properties of these functions
lead computer algebra systems into various conundrums. We discuss how
they can manipulate such functions in a useful fashion.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[DLMF]{DLMF}.
``Digital Library of Mathematical Functions''
\verbdlmf.nist.gov/software/#T1
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Bron90c,
+ author = "Bronstein, Manuel",
+ title = "On the integration of elementary functions",
+ journal = "Journal of Symbolic Computation",
+ volume = "9",
+ number = "2",
+ pages = "117173",
+ year = "1990",
+ month = "February"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dooley 99]{Doo99} Dooley, Sam editor.
ISSAC 99: July 2931, 1999, Simon Fraser University,
Vancouver, BC, Canada: proceedings of the 1999 International Symposium on
Symbolic and Algebraic Computation. ACM Press, New York, NY 10036, USA, 1999.
ISBN 1581130732 LCCN QA76.95.I57 1999
 keywords = "axiomref",
+\bibitem[Bronstein 93]{REFBS93} Bronstein, Manuel; Salvy, Bruno
+``Full partial fraction decomposition of rational functions''
+In Bronstein [Bro93] pp157160 ISBN 0897916042 LCCN QA76.95 I59 1993
+\verbwww.acm.org/pubs/citations/proceedings/issac/164081/
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dos Reis 12]{DR12} Dos Reis, Gabriel
``A System for Axiomatic Programming''
Proc. Conf. on Intelligent Computer Mathematics, Springer (2012)
\verbwww.axiomatics.org/~gdr/liz/cicm2012.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/DR12.pdf
 keywords = "axiomref",
+\bibitem[Bronstein 90]{Bro90b} Bronstein, Manuel
+``A Unification of Liouvillian Extensions''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro90b.pdf
+ abstract = "
+ We generalize Liouville's theory of elementary functions to a larger
+ class of differential extensions. Elementary, Liouvillian and
+ trigonometric extensions are all special cases of our extensions. In
+ the transcendental case, we show how the rational techniques of
+ integration theory can be applied to our extensions, and we give a
+ unified presentation which does not require separate cases for
+ different monomials."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present the design and implementation of a system for axiomatic
programming, and its application to mathematical software
construction. Key novelties include a direct support for userdefined
axioms establishing local equality between types, and overload
resolution based on equational theories and userdefined local
axioms. We illustrate uses of axioms, and their organization into
concepts, in structured generic programming as practiced in
computational mathematical systems.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Doye 97]{Doy97} Doye, Nicolas James
``Order Sorted Computer Algebra and Coercions''
Ph.D. Thesis University of Bath 1997
%\verbaxiomdeveloper.org/axiomwebsite/papers/Doy97.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@book{Bron97,
+ author = "Bronstein, Manuel",
+ title = "Symbolic Integration ITranscendental Functions",
+ publisher = "Springer, Heidelberg",
+ year = "1997",
+ isbn = "3540214933",
+ url = "http://evilwire.org/arrrXiv/Mathematics/Bronstein,_Symbolic_Integration_I,1997.pdf",
+ paper = "Bron97.pdf"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Computer algebra systems are large collections of routines for solving
mathematical problems algorithmically, efficiently and above all,
symbolically. The more advanced and rigorous computer algebra systems
(for example, Axiom) use the concept of strong types based on
ordersorted algebra and category theory to ensure that operations are
only applied to expressions when they ``make sense''.

In cases where Axiom uses notions which are not covered by current
mathematics we shall present new mathematics which will allow us to
prove that all such cases are reducible to cases covered by the
current theory. On the other hand, we shall also point out all the
cases where Axiom deviates undesirably from the mathematical ideal.
Furthermore we shall propose solutions to these deviations.
+\begin{chunk}{ignore}
+\bibitem[Bronstein 05a]{Bro05a} Bronstein, Manuel
+``The Poor Man's Integrator, a parallel integration heuristic''
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/pmint/pmint.txt
+\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/pmint/examples
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro05a.txt
Strongly typed systems (especially of mathematics) become unusable
unless the system can change the type in a way a user expects. We wish
any change expected by a user to be automated, ``natural'', and
unique. ``Coercions'' are normally viewed as ``natural type changing
maps''. This thesis shall rigorously define the word ``coercion'' in
the context of computer algebra systems.
+\end{chunk}
We shall list some assumptions so that we may prove new results so
that all coercions are unique. This concept is called ``coherence''.
+\begin{chunk}{axiom.bib}
+@article{Bron06,
+ author = "Bronstein, M.",
+ title = "Parallel integration",
+ journal = "Programming and Computer Software",
+ year = "2006",
+ issn = "03617688",
+ volume = "32",
+ number = "1",
+ doi = "10.1134/S0361768806010075",
+ url = "http://dx.doi.org/10.1134/S0361768806010075",
+ publisher = "Nauka/Interperiodica",
+ pages = "5960",
+ paper = "Bron06.pdf",
+ abstract = "
+ Parallel integration is an alternative method for symbolic
+ integration. While also based on Liouville's theorem, it handles all
+ the generators of the differential field containing the integrand ``in
+ parallel'', i.e. all at once rather than considering only the topmost
+ one in a recursive fasion. Although it still contains heuristic
+ aspects, its ease of implementation, speed, high rate of success, and
+ ability to integrate functions that cannot be handled by the Risch
+ algorithm make it an attractive alternative."
+}
We shall give an algorithm for automatically creating all coercions in
type system which adheres to a set of assumptions. We shall prove that
this is an algorithm and that it always returns a coercion when one
exists. Finally, we present a demonstration implementation of this
automated coerion algorithm in Axiom.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Doye 99]{Doy99} Doye, Nicolas J.
``Automated coercion for Axiom''
In Dooley [Doo99], pp229235
ISBN 1581130732 LCCN QA76.95.I57 1999 ACM Press
\verbwww.acm.org/citation.cfm?id=309944
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Bron07,
+ author = "Bronstein, Manuel",
+ title = "Structure theorems for parallel integration",
+ journal = "Journal of Symbolic Computation",
+ volume = "42",
+ number = "7",
+ pages = "757769",
+ year = "2007",
+ month = "July",
+ paper = "Bron07.pdf",
+ abstract = "
+ We introduce structure theorems that refine Liouville's Theorem on
+ integration in closed form for general derivations on multivariate
+ rational function fields. By predicting the arguments of the new
+ logarithms that an appear in integrals, as well as the denominator of
+ the rational part, those theorems provide theoretical backing for the
+ RischNorman integration method. They also generalize its applicability
+ to nonmonomial extensions, for example the Lambert W function."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dominguez 01]{DR01} Dom\'inguez, C\'esar; Rubio, Julio
``Modeling Inheritance as Coercion in a Symbolic Computation System''
ISSAC 2001 ACM 1581134177/01/0007
%\verbaxiomdeveloper.org/axiomwebsite/papers/DR01.pdf
 keywords = "axiomref",
+\bibitem[Charlwood 07]{Charl07} Charlwood, Kevin
+``Integration on Computer Algebra Systems''
+The Electronic J of Math. and Tech. Vol 2, No 3, ISSN 19332823
+\verb12000.org/my_notes/ten_hard_integrals/paper.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Charl07.pdf
+ abstract = "
+ In this article, we consider ten indefinite integrals and the ability
+ of three computer algebra systems (CAS) to evaluate them in
+ closedform, appealing only to the class of real, elementary
+ functions. Although these systems have been widely available for many
+ years and have undergone major enhancements in new versions, it is
+ interesting to note that there are still indefinite integrals that
+ escape the capacity of these systems to provide antiderivatves. When
+ this occurs, we consider what a user may do to find a solution with
+ the aid of a CAS."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this paper the analysis of the data structures used in a symbolic
computation system, called Kenzo, is undertaken. We deal with the
specification of the inheritance relationship since Kenzo is an
objectoriented system, written in CLOS, the Common Lisp Object
System. We focus on a particular case, namely the relationship between
simplicial sets and chain complexes, showing how the ordersorted
algebraic specifications formalisms can be adapted, through the
``inheritance as coercion'' metaphor, in order to model this Kenzo
fragment.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Dunstan 97]{Dun97} Dunstan, Martin and Ursula, Martin and Linton, Steve
``Embedded Verification Techniques for Computer Algebra Systems''
Grant citation GR/L48256 Nov 1, 1997Feb 28, 2001
\verbwww.cs.standrews.ac.uk/research/output/detail?output=ML97.php
 keywords = "axiomref",
+\bibitem[Charlwood 08]{Charl08} Charlwood, Kevin
+``Symbolic Integration Problems''
+\verbwww.apmaths.uwo.ca/~arich/IndependentTestResults/CharlwoodIntegrationProblems.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Charl08.pdf
+ abstract = "
+ A list of the 50 example integration problems from Kevin Charlwood's 2008
+ article ``Integration on Computer Algebra Systems''. Each integral along
+ with its optimal antiderivative (that is, the best antiderivative found
+ so far) is shown."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Adams 01]{DGKM01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne;
Kelsey, Tom; Martin, Ursula; Owre, Sam
``Computer Algebra meets Automated Theorem Proving: Integrating Maple and PVS''
TPHOLS 2001, Edinburgh
\verbwww.csl.sri.com/~owre/papers/tphols01/tphols01.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/DGKM01.pdf
 keywords = "axiomref",
+\bibitem[Cherry 84]{Che84} Cherry, G.W.
+``Integration in Finite Terms with Special Functions: The Error Function''
+J. Symbolic Computation (1985) Vol 1 pp283302
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Che84.pdf
+ abstract = "
+ A decision procedure for integrating a class of transcendental
+ elementary functions in terms of elementary functions and error
+ functions is described. The procedure consists of three mutually
+ exclusive cases. In the first two cases a generalised procedure for
+ completing squares is used to limit the error functions which can
+ appear in the integral of a finite number. This reduces the problem
+ to the solution of a differential equation and we use a result of
+ Risch (1969) to solve it. The third case can be reduced to the
+ determination of what we have termed $\sum$decompositions. The resutl
+ presented here is the key procuedure to a more general algorithm which
+ is described fully in Cherry (1983)."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe an interface between version 6 of the Maple computer
algebra system with the PVS automated theorem prover. The interface is
designed to allow Maple users access to the robust and checkable proof
environment of PVS. We also extend this environment by the provision
of a library of proof strategies for use in real analysis. We
demonstrate examples using the interface and the real analysis
library. These examples provide proofs which are both illustrative and
applicable to genuine symbolic computation problems.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Duval 92]{DJ92} Duval D.; Jung, F.
``Examples of problem solving using computer algebra''
IFIP Transactions. A. Computer Science and Technology, A2 pp133141, 143 1992
CODEN ITATEC. ISSN 09265473
 keywords = "axiomref",
+\bibitem[Cherry 86]{Che86} Cherry, G.W.
+``Integration in Finite Terms with Special Functions:
+The Logarithmic Integral''
+SIAM J. Comput. Vol 15 pp121 February 1986
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Duval 94]{Duv94} Duval, Dominique
``Symbolic or algebraic computation?''
Madrid Spain, NAG conference (private copy of paper)
 keywords = "axiomref",
+\bibitem[Cherry 89]{Che89} Cherry, G.W.
+``An Analysis of the Rational Exponential Integral''
+SIAM J. Computing Vol 18 pp 893905 (1989)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Che89.pdf
+ abstract = "
+ In this paper an algorithm is presented for integrating expressions of
+ the form $\int{ge^f~dx}$, where $f$ and $g$ are rational functions of
+ $x$, in terms of a class of special functions called the special
+ incomplete $\Gamma$ functions. This class of special functions
+ includes the exponential integral, the error functions, the sine and
+ cosing integrals, and the Fresnel integrals. The algorithm presented
+ here is an improvement over those published previously for integrating
+ with special functions in the following ways: (i) This algorithm
+ combines all the above special functions into one algorithm, whereas
+ previously they were treated separately, (ii) Previous algorithms
+ require that the underlying field of constants be algebraically
+ closed. This algorithm, however, works over any field of
+ characteristic zero in which the basic field operations can be carried
+ out. (iii) This algorithm does not rely on Risch's solution of the
+ differential equation $y^\prime + fy = g$. Instead, a more direct
+ method of undetermined coefficients is used."
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Duva95,
 author = "Duval, D.",
 title = "Evaluation dynamique et cl\^oture alg\'ebrique en Axiom",
 journal = "Journal of Pure and Applied Algebra",
 volume = "99",
 year = "1995",
 pages = "267295.",
 keywords = "axiomref"
}
+\begin{chunk}{ignore}
+\bibitem[Churchill 06]{Chur06} Churchill, R.C.
+``Liouville's Theorem on Integration Terms of Elementary Functions''
+\verbwww.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Chur06.pdf
+ abstract = "
+ This talk should be regarded as an elementary introduction to
+ differential algebra. It culminates in a purely algebraic proof, due
+ to M. Rosenlicht, of an 1835 theorem of Liouville on the existence of
+ ``elementary'' integrals of ``elementary'' functions. The precise
+ meaning of elementary will be specified. As an application of that
+ theorem we prove that the indefinite integral $\int{e^{x^2}}~dx$
+ cannot be expressed in terms of elementary functions.
+ \begin{itemize}
+ \item Preliminaries on Meromorphic Functions
+ \item Basic (Ordinary) Differential Algebra
+ \item Differential Ring Extensions with No New Constants
+ \item Extending Derivations
+ \item Integration in Finite Terms
+ \end{itemize}"
\end{chunk}
\subsection{E} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Erocal 10]{ES10} Er\"ocal, Burcin; Stein, William
``The Sage Project''
\verbwstein.org/papers/icms/icms_2010.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/ES10.pdf
 keywords = "axiomref",
+\bibitem[Davenport 79b]{Dav79b} Davenport, James Harold
+``On the Integration of Algebraic Functions''
+SpringerVerlag Lecture Notes in Computer Science 102
+ISBN 0387102906
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Sage is a free, open source, selfcontained distribution of
mathematical software, including a large library that provides a
unified interface to the components of this distribution. This library
also builds on the components of Sage to implement novel algorithms
covering a broad range of mathematical functionality from algebraic
combinatorics to number theory and arithmetic geometry.
\end{adjustwidth}

\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Fateman 90]{Fat90} Fateman, R. J.
``Advances and trends in the design and construction of algebraic
manipulation systems''
In Watanabe and Nagata [WN90], pp6067 ISBN 0897914015 LCCN QA76.95.I57 1990
 keywords = "axiomref",
+\bibitem[Davenport 79c]{Dav79c} Davenport, J. H.
+``Algorithms for the Integration of Algebraic Functions''
+Lecture Notes in Computer Science V 72 pp415425 (1979)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav79c.pdf
+ abstract = "
+ The problem of finding elementary integrals of algebraic functions has
+ long been recognized as difficult, and has sometimes been thought
+ insoluble. Risch stated a theorem characterising the integrands with
+ elementary integrals, and we can use the language of algebraic
+ geometry and the techniques of Davenport to yield an algorithm that will
+ always produce the integral if it exists. We explain the difficulty in
+ the way of extending this algorithm, and outline some ways of solving
+ it. Using work of Manin we are able to solve the problem in all cases
+ where the algebraic expressions depend on a parameter as well as on
+ the variable of integration."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fateman 05]{Fat05} Fateman, R. J.
``An incremental approach to building a mathematical expert out of software''
4/19/2005\hfill
\verbwww.cs.berkeley.edu/~fateman/papers/axiom.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat05.pdf
 keywords = "axiomref",
+\bibitem[Davenport 82a]{Dav82a} Davenport, J.H.
+``The Parallel Risch Algorithm (I)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav82a.pdf
+ abstract = "
+ In this paper we review the socalled ``parallel Risch'' algorithm for
+ the integration of transcendental functions, and explain what the
+ problems with it are. We prove a positive result in the case of
+ logarithmic integrands."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fateman 06]{Fat06} Fateman, R. J.
``Building Algebra Systems by Overloading Lisp''
\verbwww.cs.berkeley.edu/~fateman/generic/overloadsmall.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat06.pdf
 keywords = "axiomref",
+\bibitem[Davenport 82]{Dav82} Davenport, J.H.
+``On the Parallel Risch Algorithm (III): Use of Tangents''
+SIGSAM V16 no. 3 pp36 August 1982
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Some of the earliest computer algebra systems (CAS) looked like
overloaded languages of the same era. FORMAC, PL/I FORMAC, Formula
Algol, and others each took advantage of a preexisting language base
and expanded the notion of a numeric value to include mathematical
expressions. Much more recently, perhaps encouraged by the growth in
popularity of C++, we have seen a renewal of the use of overloading to
implement a CAS.

This paper makes three points. 1. It is easy to do overloading in
Common Lisp, and show how to do it in detail. 2. Overloading per se
provides an easy solution to some simple programming problems. We show
how it can be used for a ``demonstration'' CAS. Other simple and
plausible overloadings interact nicely with this basic system. 3. Not
all goes so smoothly: we can view overloading as a case study and
perhaps an object lesson since it fails to solve a number of
fairlywell articulated and difficult design issues in CAS for which
other approaches are preferable.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Faure 00a]{FDN00a} Faure, Christ\'ele; Davenport, James
``Parameters in Computer Algebra''
 keywords = "axiomref",
+\bibitem[Davenport 03]{Dav03} Davenport, James H.
+``The Difficulties of Definite Integration''
+\verbwww.researchgate.net/publication/
+\verb247837653_The_Diculties_of_Definite_Integration/file/72e7e52a9b1f06e196.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav03.pdf
+ abstract = "
+ Indefinite integration is the inverse operation to differentiation,
+ and, before we can understand what we mean by indefinite integration,
+ we need to understand what we mean by differentiation."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Faure 00b]{FDN00b} Faure, Christ\'ele; Davenport, James;
Naciri, Hanane
``Multivalues Computer Algebra''
ISSN 02496399 Institut National De Recherche en Informatique et en
Automatique Sept. 2000 No. 4001
\verbhal.inria.fr/inria00072643/PDF/RR4401.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/FDN00b.pdf
 keywords = "axiomref",
+\bibitem[Fateman 02]{Fat02} Fateman, Richard
+``Symbolic Integration''
+\verbinst.eecs.berkeley.edu/~cs282/sp02/lects/14.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat02.pdf
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
One of the main strengths of computer algebra is being able to solve a
family of problems with one computation. In order to express not only
one problem but a family of problems, one introduces some symbols
which are in fact the parameters common to all the problems of the
family.
+\begin{chunk}{axiom.bib}
+@inproceedings{Gedd89,
+ author = "Geddes, K. O. and Stefanus, L. Y.",
+ title = "On the Rischnorman Integration Method and Its Implementation
+ in MAPLE",
+ booktitle = "Proc. of the ACMSIGSAM 1989 Int. Symp. on Symbolic and
+ Algebraic Computation",
+ series = "ISSAC '89",
+ year = "1989",
+ isbn = "0897913256",
+ location = "Portland, Oregon, USA",
+ pages = "212217",
+ numpages = "6",
+ url = "http://doi.acm.org/10.1145/74540.74567",
+ doi = "10.1145/74540.74567",
+ acmid = "74567",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ paper = "Gedd89.pdf",
+ abstract = "
+ Unlike the Recursive Risch Algorithm for the integration of
+ transcendental elementary functions, the RischNorman Method processes
+ the tower of field extensions directly in one step. In addition to
+ logarithmic and exponential field extensions, this method can handle
+ extentions in terms of tangents. Consequently, it allows trigonometric
+ functions to be treated without converting them to complex exponential
+ form. We review this method and describe its implementation in
+ MAPLE. A heuristic enhancement to this method is also presented."
+}
The user must be able to understand in which way these parameters
affect the result when he looks at the answer. Otherwise it may lead
to completely wrong calculations, which when used for numerical
applications bring nonsensical answers. This is the case in most
current Computer Algebra Systems we know because the form of the
answer is never explicitly conditioned by the values of the
parameters. The user is not even informed that the given answer may be
wrong in some cases then computer algebra systems can not be entirely
trustworthy. We have introduced multivalued expressions called {\sl
conditional} expressions, in which each potential value is associated
with a condition on some parameters. This is used, in particular, to
capture the situation in integration, where the form of the answer can
depend on whether certain quantities are positive, negative or
zero. We show that it is also necessary when solving modular linear
equations or deducing congruence conditions from complex expressions.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fitch 84]{Fit84} Fitch, J. P. (ed)
EUROSAM '84: International Symposium on Symbolic and
Algebraic Computation, Cambridge, England, July 911, 1984, volume 174 of
Lecture Notes in Computer Science. SpringerVerlag, Berlin, Germany /
Heildelberg, Germany / London, UK / etc., 1984 ISBN 038713350X
LCCN QA155.7.E4 I57 1984
 keywords = "axiomref",
+\bibitem[Geddes 92a]{GCL92a} Geddes, K.O.; Czapor, S.R.; Labahn, G.
+``The Risch Integration Algorithm''
+Algorithms for Computer Algebra, Ch 12 pp511573 (1992)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/GCL92a.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fitch 93]{Fit93} Fitch, J. (ed)
Design and Implementation of Symbolic Computation Systems
International Symposium DISCO '92 Proceedings. SpringerVerlag, Berlin,
Germany / Heildelberg, Germany / London, UK / etc., 1993. ISBN 0387572724
(New York), 3540572724 (Berlin). LCCN QA76.9.S88I576 1992
 keywords = "axiomref",
+\bibitem[Hardy 1916]{Hard16} Hardy, G.H.
+``The Integration of Functions of a Single Variable''
+Cambridge Unversity Press, Cambridge, 1916
+% REF:00002
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fogus 11]{Fog11} Fogus, Michael
``UnConj''
\verbclojure.com/blog/2011/11/22/unconj.html
 keywords = "axiomref",
+\bibitem[Harrington 78]{Harr87} Harrington, S.J.
+``A new symbolic integration system in reduce''
+\verbcomjnl.oxfordjournals.or/content/22/2/127.full.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Harr87.pdf
+ abstract = "
+ A new integration system, employing both algorithmic and pattern match
+ integration schemes is presented. The organization of the system
+ differs from that of earlier programs in its emphasis on the
+ algorithmic approach to integration, its modularity and its ease of
+ revision. The new NormanRish algorithm and its implementation at the
+ University of Cambridge are employed, supplemented by a powerful
+ collection of simplification and transformation rules. The facility
+ for user defined integrals and functions is also included. The program
+ is both fast and powerful, and can be easily modified to incorporate
+ anticipated developments in symbolic integration."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fortenbacher 90]{For90} Fortenbacher, A.
``Efficient type inference and coercion in computer algebra''
In Miola [Mio90], pp5660. ISBN 0387525319 (New York), 3540525319
(Berlin). LCCN QA76.9.S88I576 1990
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Herm1872,
+ author = "Hermite, E.",
+ title = "Sur l'int\'{e}gration des fractions rationelles",
+ journal = "Nouvelles Annales de Math\'{e}matiques",
+ volume = "11",
+ pages = "145148",
+ year = "1872"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fouche 90]{Fou90} Fouche, Francois
``Une implantation de l'algorithme de Kovacic en Scratchpad''
Technical report, Institut de Recherche Math{\'{e}}matique Avanc{\'{e}}e''
Strasbourg, France, 1990 31pp
 keywords = "axiomref",
+\bibitem[Horowitz 71]{Horo71} Horowitz, Ellis
+``Algorithms for Partial Fraction Decomposition and Rational Function
+ Integration''
+SYMSAC '71 Proc. ACM Symp. on Symbolic and Algebraic Manipulation (1971)
+pp441457
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Horo71.pdf REF:00018
+ abstract = "
+ Algorithms for symbolic partial fraction decomposition and indefinite
+ integration of rational functions are described. Two types of
+ partial fraction decomposition are investigated, squarefree and
+ complete squarefree. A method is derived, based on the solution of
+ a linear system, which produces the squarefree decomposition of any
+ rational function, say A/B. The computing time is show to be
+ $O(n^4(ln nf)^2)$ where ${\rm deg}(A) < {\rm\ deg}(B) = n$ and $f$
+ is a number which is closely related to the size of the coefficients
+ which occur in A and B. The complete squarefree partical fraction
+ decomposition can then be directly obtained and it is shown that the
+ computing time for this process is also bounded by $O(n^4(ln nf)^2)$."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[FSF 14]{FSF14} FSF
``Free Software Directory''
\verbdirectory.fsf.org/wiki/Axiom
 keywords = "axiomref",
+\bibitem[Jeffrey 97]{Jeff97} Jeffrey, D.J.; Rich, A.D.
+``Recursive integration of piecewisecontinuous functions''
+\verbwww.cybertester.com/data/recint.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Jeff97.pdf
+ abstract = "
+ An algorithm is given for the integration of a class of
+ piecewisecontinuous functions. The integration is with respect to a
+ real variable, because the functions considered do not in general
+ allow integration in the complex plane to be defined. The class of
+ integrands includes commonly occurring waveforms, such as square
+ waves, triangular waves, and the floor function; it also includes the
+ signum function. The algorithm can be implemented recursively, and it
+ has the property of ensuring that integrals are continuous on domains
+ of maximum extent."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Frisco ]{Fris} Frisco
``Objectives and Results''
\verbwww.nag.co.uk/projects/frisco/frisco/node3.htm
 keywords = "axiomref",
+\bibitem[Jeffrey 99]{Jeff99} Jeffrey, D.J.; Labahn, G.; Mohrenschildt, M.v.;
+Rich, A.D.
+``Integration of the signum, piecewise and related functions''
+\verbcs.uwaterloo.ca/~glabahn/Papers/issac992.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Jeff99.pdf
+ abstract = "
+ When a computer algebra system has an assumption facility, it is
+ possible to distinguish between integration problems with respect to a
+ real variable, and those with respect to a complex variable. Here, a
+ class of integration problems is defined in which the integrand
+ consists of compositions of continuous functions and signum functions,
+ and integration is with respect to a real variable. Algorithms are
+ given for evaluating such integrals."
\end{chunk}
\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Gebauer 86]{GM86} Gebauer, R{\"u}diger; M{\"o}ller, H. Michael
``Buchberger's algorithm and staggered linear bases''
In Bruce W. Char, editor. Proceedings of the 1986
Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 2123, 1986
Waterloo, Ontario, pp218221 ACM Press, New York, NY 10036, USA, 1986.
ISBN 0897911997 LCCN QA155.7.E4 A281 1986 ACM order number 505860
 keywords = "axiomref",
+\bibitem[Kiymaz 04]{Kiym04} Kiymaz, Onur; Mirasyedioglu, Seref
+``A new symbolic computation for formal integration with exact power series''
+%\verbaxiomdeveloper.org/axiomwebsite/Kiym04.pdf
+ abstract = "
+ This paper describes a new symbolic algorithm for formal integration
+ of a class of functions in the context of exact power series by using
+ generalized hypergeometric series and computer algebraic technique."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gebauer 88]{GM88} Gebauer, R.; M{\"o}ller, H. M.
``On an installation of Buchberger's algorithm''
Journal of Symbolic Computation, 6(23) pp275286 1988
CODEN JSYCEH ISSN 07477171
\verbwww.sciencedirect.com/science/article/pii/S0747717188800488/pdf
\verb?md5=f6ccf63002ef3bc58aaa92e12ef18980&
\verbpid=1s2.0S0747717188800488main.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/GM88.pdf
 keywords = "axiomref",
+\bibitem[Knowles 93]{Know93} Knowles, P.
+``Integration of a class of transcendental liouvillian
+functions with errorfunctions i''
+Journal of Symbolic Computation Vol 13 pp525543 (1993)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Buchberger's algorithm calculates Groebner bases of polynomial
ideals. Its efficiency depends strongly on practical criteria for
detecting superfluous reductions. Buchberger recommends two
criteria. The more important one is interpreted in this paper as a
criterion for detecting redundant elements in a basis of a module of
syzygies. We present a method for obtaining a reduced, nearly minimal
basis of that module. The simple procedure for detecting (redundant
syzygies and )superfluous reductions is incorporated now in our
installation of Buchberger's algorithm in SCRATCHPAD II and REDUCE
3.3. The paper concludes with statistics stressing the good
computational properties of these installations.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Knowles 95]{Know95} Knowles, P.
+``Integration of a class of transcendental liouvillian
+functions with errorfunctions ii''
+Journal of Symbolic Computation Vol 16 pp227241 (1995)
+
+\end{chunk}
\begin{chunk}{axiom.bib}
@book{Gedd92,
 author = "Geddes, Keith and Czapor, O. and Stephen R. and Labahn, George",
 title = "Algorithms For Computer Algebra",
 publisher = "Kluwer Academic Publishers",
 isbn = "0792392590",
 month = "September",
 year = "1992",
 keywords = "axiomref"
+@article{Krag09,
+ author = "Kragler, R.",
+ title = "On Mathematica Program for Poor Man's Integrator Algorithm",
+ journal = "Programming and Computer Software",
+ volume = "35",
+ number = "2",
+ pages = "6378",
+ year = "2009",
+ issn = "03617688",
+ paper = "Krag09.pdf",
+ abstract = "
+ In this paper by means of computer experiment we study advantages and
+ disadvantages of the heuristical method of ``parallel integrator''. For
+ this purpose we describe and use implementation of the method in
+ Mathematica. In some cases we compare this implementation with the original
+ one in Maple."
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gianni 87]{Gia87} Gianni, Patrizia
``Primary Decomposition of Ideals''
in [Wit87], pp1213
 keywords = "axiomref",
+\bibitem[Lang 93]{Lang93} Lang, S.
+``Algebra''
+AddisonWesly, New York, 3rd edition 1993
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gianni 88]{Gia88} Gianni, Patrizia.; Trager, Barry.;
Zacharias, Gail.
``Gr\"obner Bases and Primary Decomposition of Polynomial Ideals''
J. Symbolic Computation 6, 149167 (1988)
\verbwww.sciencedirect.com/science/article/pii/S0747717188800403/pdf
\verb?md5=40c29b67947035884904fd4597ddf710&
\verbpid=1s2.0S0747717188800403main.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Gia88.pdf
 keywords = "axiomref",
+\bibitem[Leerawat 02]{Leer02} Leerawat, Utsanee; Laohakosol, Vichian
+``A Generalization of Liouville's Theorem on Integration in Finite Terms''
+\verbwww.mathnet.or.kr/mathnet/kms_tex/113666.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Leer02.pdf
+ abstract = "
+ A generalization of Liouville's theorem on integration in finite
+ terms, by enlarging the class of fields to an extension called
+ EiGamma extension is established. This extension includes the
+ $\mathcal{E}\mathcal{L}$elementary extensions of Singer, Saunders and
+ Caviness and contains the Gamma function."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gianni 89a]{Gia89} Gianni, P. (Patrizia) (ed)
Symbolic and Algebraic Computation.
International Symposium ISSAC '88, Rome, Italy, July 48, 1988. Proceedings,
volume 358 of Lecture Notes in Computer Science. SpringerVerlag, Berlin,
Germany / Heildelberg, Germany / London, UK / etc., 1989. ISBN 3540510842
LCCN QA76.95.I57 1988 Conference held jointly with AAECC6
 keywords = "axiomref",
+\bibitem[Leslie 09]{Lesl09} Leslie, Martin
+``Why you can't integrate exp($x^2$)''
+\verbmath.arizona.edu/~mleslie/files/integrationtalk.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Lesl09.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gianni 89b]{GM89} Gianni, P.; Mora, T.
``Algebraic solution of systems of polynomial equations using
Gr{\"o}bner bases.''
In Huguet and Poli [HP89], pp247257 ISBN 3540510826 LCCN QA268.A35 1987
 keywords = "axiomref",
+\bibitem[Lichtblau 11]{Lich11} Lichtblau, Daniel
+``Symbolic definite (and indefinite) integration: methods and open issues''
+ACM Comm. in Computer Algebra Issue 175, Vol 45, No.1 (2011)
+\verbwww.sigsam.org/bulletin/articles/175/issue175.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Lich11.pdf
+ abstract = "
+ The computation of definite integrals presents one with a variety of
+ choices. There are various methods such as NewtonLeibniz or Slater's
+ convolution method. There are questions such as whether to split or
+ merge sums, how to search for singularities on the path of
+ integration, when to issue conditional results, how to assess
+ (possibly conditional) convergence, and more. These various
+ considerations moreover interact with one another in a multitude of
+ ways. Herein we discuss these various issues and illustrate with examples."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gil 92]{Gil92} Gil, I.
``Computation of the Jordan canonical form of a square matrix (using
the Axiom programming language)''
In Wang [Wan92], pp138145.
ISBN 0897914899 (soft cover), 0897914902 (hard cover)
LCCN QA76.95.I59 1992
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Liou1833a,
+ author = "Liouville, Joseph",
+ title = "Premier m\'{e}moire sur la d\'{e}termination des int\'{e}grales
+ dont la valeur est alg\'{e}brique",
+ journal = "Journal de l'Ecole Polytechnique",
+ volume = "14",
+ pages = "124128",
+ year = "1833"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[GomezDiaz 92]{Gom92} G\'omezD'iaz, Teresa
``Quelques applications de l`\'evaluation dynamique''
Ph.D. Thesis L'Universite De Limoges March 1992
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Liou1833b,
+ author = "Liouville, Joseph",
+ title = "Second m\'{e}moire sur la d\'{e}termination des int\'{e}grales
+ dont la valeur est alg\'{e}brique",
+ journal = "Journal de l'Ecole Polytechnique",
+ volume = "14",
+ pages = "149193",
+ year = "1833"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[GomezDiaz 93]{Gom93} G\'omezD\'iaz, Teresa
``Examples of using Dynamic Constructible Closure''
IMACS Symposium SC1993
%\verbaxiomdeveloper.org/axiomwebsite/papers/Gom93.pdf
 keywords = "axiomref",
+\bibitem[Liouville 1833c]{Lio1833c} Liouville, Joseph
+``Note sur la determination des int\'egrales dont la
+valeur est alg\'ebrique''
+Journal f\"ur die Reine und Angewandte Mathematik,
+Vol 10 pp 247259, (1833)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present here some examples of using the ``Dynamic Constructible
Closure'' program, which performs automatic case distinction in
computations involving parameters over a base field $K$. This program
is an application of the ``Dynamic Evaluation'' principle, which
generalizes traditional evaluation and was first used to deal with
algebraic numbers.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Goodwin 91]{GBL91} Goodwin, B. M.; Buonopane, R. A.; Lee, A.
``Using MathCAD in teaching material and energy balance concepts''
In Anonymous [Ano91], pp345349 (vol. 1) 2 vols.
 keywords = "axiomref",
+\bibitem[Liouville 1833d]{Lio1833d} Liouville, Joseph
+``Sur la determination des int\'egrales dont la valeur est
+alg\'ebrique''
+{\sl Journal de l'Ecole Polytechnique}, 14:124193, 1833
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Golden 4]{GH84} Golden, V. Ellen; Hussain, M. A. (eds)
Proceedings of the 1984 MACSYMA Users' Conference:
Schenectady, New York, July 2325, 1984, General Electric,
Schenectady, NY, USA, 1984
 keywords = "axiomref",
+\bibitem[Liouville 1835]{Lio1835} Liouville, Joseph
+``M\'emoire sur l'int\'gration d'une classe de fonctions
+transcendentes''
+Journal f\"ur die Reine und Angewandte Mathematik,
+Vol 13(2) pp 93118, (1835)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gonnet 96]{Gon96} Gonnet, Gaston H.
``Official verion 1.0 of the Meta Content Dictionary''
\verbwww.inf.ethz.ch/personal/gonnet/ContDict/Meta
 keywords = "axiomref",
+\bibitem[Marc 94]{Marc94} Marchisotto, Elena Anne; Zakeri, GholemAll
+``An Invitation to Integration in Finite Terms''
+College Mathematics Journal Vol 25 No 4 (1994) pp295308
+\verbwww.rangevoting.org/MarchisottoZint.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Marc94.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Goodloe 93]{GL93} Goodloe, A.; Loustaunau, P.
``An abstract data type development of graded rings''
In Fitch [Fit93], pp193202. ISBN 0387572724 (New York),
3540572724 (Berlin). LCCN QA76.9.S88I576 1992
 keywords = "axiomref",
+\bibitem[Marik 91]{Mari91} Marik, Jan
+``A note on integration of rational functions''
+\verbdml.cz/bitstream/handle/10338.dmlcz/126024/MathBohem_11619914_9.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Mari91.pdf
+ abstract = "
+ Let $P$ and $Q$ be polynomials in one variable with complex coefficients
+ and let $n$ be a natural number. Suppose that $Q$ is not constant and
+ has only simple roots. Then there is a rational function $\varphi$
+ with $\varphi^\prime=P/Q^{n+1}$ if and only if the Wronskian of the
+ functions $Q^\prime$, $(Q^2)^\prime,\ldots\,(Q^n)^\prime$,$P$ is
+ divisible by $Q$."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gottliebsen 05]{GKM05} Gottliebsen, Hanne; Kelsey, Tom;
Martin, Ursula
``Hidden verification for computational mathematics''
Journal of Symbolic Computation, Vol39, Num 5, pp539567 (2005)
\verbwww.sciencedirect.com/science/article/pii/S0747717105000295
%\verbaxiomdeveloper.org/axiomwebsite/papers/GKM05.pdf
 keywords = "axiomref",
+\bibitem[Moses 76]{Mos76} Moses, Joel
+``An introduction to the Risch Integration Algorithm''
+ACM Proc. 1976 annual conference pp425428
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Mos76.pdf REF:00048
+ abstract = "
+ Risch's decision procedure for determining the integrability in closed
+ form of the elementary functions of the calculus is presented via
+ examples. The exponential and logarithmic cases of the algorithsm had
+ been implemented for the MACSYMA system several years ago. The
+ implementation of the algebraic case of the algorithm is the subject
+ of current research."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present hidden verification as a means to make the power of
computational logic available to users of computer algebra systems
while shielding them from its complexity. We have implemented in PVS a
library of facts about elementary and transcendental function, and
automatic procedures to attempt proofs of continuity, convergence and
differentiability for functions in this class. These are called
directly from Maple by a simple pipelined interface. Hence we are
able to support the analysis of differential equations in Maple by
direct calls to PVS for: result refinement and verification, discharge
of verification conditions, harnesses to ensure more reliable
differential equation solvers, and verifiable lookup tables.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Grabe 98]{Gra98} Gr\"abe, HansGert
``About the Polynomial System Solve Facility of Axiom, Macyma, Maple
Mathematica, MuPAD, and Reduce''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Gra98.pdf
 keywords = "axiomref",
+\bibitem[Moses 71a]{Mos71a} Moses, Joel
+``Symbolic Integration: The Stormy Decade''
+CACM Aug 1971 Vol 14 No 8 pp548560
+\verbwwwinst.eecs.berkeley.edu/~cs282/sp02/readings/mosesint.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Mos71a.pdf REF:00017
+ abstract = "
+ Three approaches to symbolic integration in the 1960's are
+ described. The first, from artificial intelligence, led to Slagle's
+ SAINT and to a large degree to Moses' SIN. The second, from algebraic
+ manipulation, led to Monove's implementation and to Horowitz' and
+ Tobey's reexamination of the Hermite algorithm for integrating
+ rational functions. The third, from mathematics, led to Richardson's
+ proof of the unsolvability of the problem for a class of functions and
+ for Risch's decision procedure for the elementary functions.
+ Generalizations of Risch's algorithm to a class of special
+ functions and programs for solving differential equations and for
+ finding the definite integral are also described."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Grabmeier 91]{GHK91} Grabmeier, J.; Huber, K.; Krieger, U.
``Das ComputeralgebraSystem AXIOM bei kryptologischen und
verkehrstheoretischen Untersuchungen des
Forschunginstituts der Deutschen Bundespost TELEKOM''
Technischer Report TR 75.91.20, IBM Wissenschaftliches
Zentrum, Heidelberg, Germany, 1991
 keywords = "axiomref",
+\bibitem[Norman 79]{Nor79} Norman, A.C.; Davenport, J.H.
+``Symbolic Integration  The Dust Settles?''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Nor79.pdf
+ abstract = "
+ By the end of the 1960s it had been shown that a computer could find
+ indefinite integrals with a competence exceeding that of typical
+ undergraduates. This practical advance was backed up by algorithmic
+ interpretations of a number of clasical results on integration, and by
+ some significant mathematical extensions to these same results. At
+ that time it would have been possible to claim that all the major
+ barriers in the way of a complete system for automated analysis had
+ been breached. In this paper we survey the work that has grown out of
+ the abovementioned early results, showing where the development has
+ been smooth and where it has spurred work in seemingly unrelated fields."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Grabmeier 92]{GS92} Grabmeier, J.; Scheerhorn, A.
``Finite fields in Axiom''
AXIOM Technical Report TR7/92 (ATR/5)(NP2522),
Numerical Algorithms Group, Inc., Downer's
Grove, IL, USA and Oxford, UK, 1992
\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
and Technical Report, IBM Heidelberg Scientific Center, 1992
 keywords = "axiomref",
+\bibitem[Ostrowski 46]{Ost46} Ostrowski, A.
+``Sur l'int\'egrabilit\'e \'el\'ementaire de quelques classes
+d'expressions''
+Comm. Math. Helv., Vol 18 pp 283308, (1946)
+% REF:00008
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Grabmeier 03]{GKW03} Grabmeier, Johannes; Kaltofen, Erich;
Weispfenning, Volker (eds)
Computer algebra handbook: foundations, applications, systems.
SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
2003. ISBN 3540654666 637pp Includes CDROM
\verbwww.springer.com/sgw/cda/frontpage/
\verb0,11855,11022214778710,00.html
 keywords = "axiomref",
+\bibitem[Raab 12]{Raab12} Raab, Clemens G.
+``Definite Integration in Differential Fields''
+\verbwww.risc.jku.at/publications/download/risc_4583/PhD_CGR.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Raab12.pdf
+ abstract = "
+ The general goal of this thesis is to investigate and develop computer
+ algebra tools for the simplification resp. evaluation of definite
+ integrals. One way of finding the value of a def inite integral is
+ via the evaluation of an antiderivative of the integrand. In the
+ nineteenth century Joseph Liouville was among the first who analyzed
+ the structure of elementary antiderivatives of elementary functions
+ systematically. In the early twentieth century the algebraic structure
+ of differential fields was introduced for modeling the differential
+ properties of functions. Using this framework Robert H. Risch
+ published a complete algorithm for transcendental elementary
+ integrands in 1969. Since then this result has been extended to
+ certain other classes of integrands as well by Michael F. Singer,
+ Manuel Bronstein, and several others. On the other hand, if no
+ antiderivative of suitable form is available, then linear relations
+ that are satisfied by the parameter integral of interest may be found
+ based on the principle of parametric integration (often called
+ differentiating under the integral sign or creative telescoping).
+
+ The main result of this thesis extends the results mentioned above to
+ a complete algo rithm for parametric elementary integration for a
+ certain class of integrands covering a majority of the special
+ functions appearing in practice such as orthogonal polynomials,
+ polylogarithms, Bessel functions, etc. A general framework is provided
+ to model those functions in terms of suitable differential fields. If
+ the integrand is Liouvillian, then the present algorithm considerably
+ improves the efficiency of the corresponding algorithm given by Singer
+ et al. in 1985. Additionally, a generalization of Czichowskiâ€™s
+ algorithm for computing the logarithmic part of the integral is
+ presented. Moreover, also partial generalizations to include other
+ types of integrands are treated.
+
+ As subproblems of the integration algorithm one also has to find
+ solutions of linear or dinary differential equations of a certain
+ type. Some contributions are also made to solve those problems in our
+ setting, where the results directly dealing with systems of
+ differential equations have been joint work with Moulay A. Barkatou.
+
+ For the case of Liouvillian integrands we implemented the algorithm in
+ form of our Mathematica package Integrator. Parts of the
+ implementation also deal with more general functions. Our procedures
+ can be applied to a significant amount of the entries in integral
+ tables, both indefinite and definite integrals. In addition, our
+ procedures have been successfully applied to interesting examples of
+ integrals that do not appear in these tables or for which current
+ standard computer algebra systems like Mathematica or Maple do not
+ succeed. We also give examples of how parameter integrals coming from
+ the work of other researchers can be solved with the software, e.g.,
+ an integral arising in analyzing the entropy of certain processes."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Griesmer 71]{GJ71} Griesmer, J. H.; Jenks, R.D.
``SCRATCHPAD/1  an interactive facility for symbolic mathematics''
In Petrick [Pet71], pp4258. LCCN QA76.5.S94 1971
\verbdelivery.acm.org/10.1145/810000/806266/p42griesmer.pdf
SYMSAC'71 Proc. second ACM Symposium on Symbolic and Algebraic
Manipulation pp4548
%\verbaxiomdeveloper.org/axiomwebsite/papers/GJ71.pdf REF:00027
 keywords = "axiomref",
+\bibitem[Raab 13]{Raab13} Raab, Clemens G.
+``Generalization of Risch's Algorithm to Special Functions''
+\verbarxiv.org/pdf/1305.1481
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Raab13.pdf
+ abstract = "
+ Symbolic integration deals with the evaluation of integrals in closed
+ form. We present an overview of Risch's algorithm including recent
+ developments. The algorithms discussed are suited for both indefinite
+ and definite integration. They can also be used to compute linear
+ relations among integrals and to find identities for special functions
+ given by parameter integrals. The aim of this presentation is twofold:
+ to introduce the reader to some basic idea of differential algebra in
+ the context of integration and to raise awareness in the physics
+ community of computer algebra algorithms for indefinite and definite
+ integration."
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Griesmer 72a]{GJ72a} Griesmer, J.; Jenks, R.
``Experience with an online symbolic math system SCRATCHPAD''
in Online'72 [Onl72] ISBN 0903796023 LCCN QA76.55.O54 1972 Two volumes
 keywords = "axiomref",
+\bibitem[Raab xx]{Raabxx} Raab, Clemens G.
+``Integration in finite terms for Liouvillian functions''
+\verbwww.mmrc.iss.ac.cn/~dart4/posters/Raab.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Raabxx.pdf
+ abstract = "
+ Computing integrals is a common task in many areas of science,
+ antiderivatives are one way to accomplish this. The problem of
+ integration in finite terms can be states as follows. Given a
+ differential field $(F,D)$ and $f \in F$, compute $g$ in some
+ elementary extension of $(F,D)$ such that $Dg = f$ if such a $g$
+ exists.
+
+ This problem has been solved for various classes of fields $F$. For
+ rational functions $(C(x), \frac{d}{dx})$ such a $g$ always exists and
+ algorithms to compute it are known already for a long time. In 1969
+ Risch published an algorithm that solves this problem when $(F,D)$ is
+ a transcendental elementary extension of $(C(x),\frac{d}{dx})$. Later
+ this has been extended towards integrands being Liouvillian functions
+ by Singer et. al. via the use of regular logexplicit extensions of
+ $(C(x),\frac{d}{dx})$. Our algorithm extends this to handling
+ transcendental Liouvillian extensions $(F,D)$ of $(C,0)$ directly
+ without the need to embed them into logexplicit extensions. For
+ example, this means that
+ \[\int{(zx)x^{z1}e^{x}dx} = x^ze^{x}\]
+ can be computed without including log(x) in the differential field."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Griesmer 72b]{GJ72b} Griesmer, James H.; Jenks, Richard D.
``SCRATCHPAD: A capsule view''
ACM SIGPLAN Notices, 7(10) pp93102, 1972. Proceedings of the symposium
on Twodimensional manmachine communications. Mark B. Wells and
James B. Morris (eds.).
 keywords = "axiomref",
+\bibitem[Rich 09]{Rich09} Rich, A.D.; Jeffrey, D.J.
+``A Knowledge Repository for Indefinite Integration Based on Transformation Rules''
+\verbwww.apmaths.uwo.ca/~arich/A%2520Rulebased%2520Knowedge%2520Repository.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Rich09.pdf
+ abstract = "
+ Taking the specific problem domain of indefinite integration, we
+ describe the ongoing development of a repository of mathematical
+ knowledge based on transformation rules. It is important that the
+ repository be not confused with a lookup table. The database of
+ transformation rules is at present encoded in Mathematica, but this is
+ only one convenient form of the repository, and it could be readily
+ translated into other formats. The principles upon which the set of
+ rules is compiled is described. One important principle is
+ minimality. The benefits of the approach are illustrated with
+ examples, and with the results of comparisons with other approaches."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Griesmer 75]{GJY75} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y
``SCRATCHPAD User's Manual''
IBM Research Publication RA70 June 1975
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@techreport{Risc68,
+ author = "Risch, Robert",
+ title = "On the integration of elementary functions which are built up
+ using algebraic operations",
+ type = "Research Report",
+ number = "SP2801/002/00",
+ institution = "System Development Corporation, Santa Monica, CA, USA",
+ year = "1968"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Griesmer 76]{GJY76} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y
``A Set of SCRATCHPAD Examples''
April 1976 (private copy)
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@techreport{Risc69a,
+ author = "Risch, Robert",
+ title = "Further results on elementary functions",
+ type = "Research Report",
+ number = "RC2042",
+ institution = "IBM Research, Yorktown Heights, NY, USA",
+ year = "1969"
+
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gruntz 94]{GM94} Gruntz, D.; Monagan, M.
``Introduction to Gauss''
SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic
Manipulation), 28(3) pp319 August 1994 CODEN SIGSBZ ISSN 01635824
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Risc69b,
+ author = "Risch, Robert",
+ title = "The problem of integration in finite terms",
+ journal = "Transactions of the American Mathematical Society",
+ volume = "139",
+ year = "1969",
+ pages = "167189",
+ paper = "Ris69b.pdf",
+ abstract = "This paper deals with the problem of telling whether a
+ given elementary function, in the sense of analysis, has an elementary
+ indefinite integral."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gruntz 96]{Gru96} Gruntz, Dominik
``On Computing Limits in a Symbolic Manipulation System''
Thesis, Swiss Federal Institute of Technology Z\"urich 1996
Diss. ETH No. 11432
\verbwww.cybertester.com/data/gruntz.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Gru96.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Risc70,
+ author = "Risch, Robert",
+ title = "The Solution of the Problem of Integration in Finite Terms",
+ journal = "Bull. AMS",
+ year = "1970",
+ issn = "00029904",
+ volume = "76",
+ number = "3",
+ pages = "605609",
+ paper = "Risc70.pdf",
+ abstract = "
+ The problem of integration in finite terms asks for an algorithm for
+ deciding whether an elementary function has an elementary indefinite
+ integral and for finding the integral if it does. ``Elementary'' is
+ used here to denote those functions build up from the rational
+ functions using only exponentiation, logarithms, trigonometric,
+ inverse trigonometric and algebraic operations. This vaguely worded
+ question has several precise, but inequivalent formulations. The
+ writer has devised an algorithm which solves the classical problem of
+ Liouville. A complete account is planned for a future publication. The
+ present note is intended to indiciate some of the ideas and techniques
+ involved."
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This thesis presents an algorithm for computing (onesided) limits
within a symbolic manipulation system. Computing limtis is an
important facility, as limits are used both by other functions such as
the definite integrator and to get directly some qualitative
information about a given function.

The algorithm we present is very compact, easy to understand and easy
to implement. It overcomes the cancellation problem other algorithms
suffer from. These goals were achieved using a uniform method, namely
by expanding the whole function into a series in terms of its most
rapidly varying subexpression instead of a recursive bottom up
expansion of the function. In the latter approach exact error terms
have to be kept with each approximation in order to resolve the
cancellation problem, and this may lead to an intermediate expression
swell. Our algorithm avoids this problem and is thus suited to be
implemented in a symbolic manipulation system.
\end{adjustwidth}
+\begin{chunk}{axiom.bib}
+@article{Risc79,
+ author = "Risch, Robert",
+ title = "Algebraic properties of the elementary functions of analysis",
+ journal = "American Journal of Mathematics",
+ volume = "101",
+ pages = "743759",
+ year = "1979"
+}
\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Boyle 88]{Boyl88} Boyle, Ann
``Future Directions for Research in Symbolic Computation''
Soc. for Industrial and Applied Mathematics, Philadelphia (1990)
\verbwww.eecis.udel.edu/~caviness/wsreport.pdf
%\verbaxiomdeveloper.org/axiomwebsite/Boyl88.pdf
 keywords = "axiomref",
+\bibitem[Ritt 48]{Ritt48} Ritt, J.F.
+``Integration in Finite Terms''
+Columbia University Press, New York 1948
+% REF:00046
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Hassner 87]{HBW87} Hassner, Martin; Burge, William H.;
Watt, Stephen M.
``Construction of Algebraic Error Control Codes (ECC) on the Elliptic
Riemann Surface''
in [Wit87], pp58
 keywords = "axiomref",
+\bibitem[Rosenlicht 68]{Ro68} Rosenlicht, Maxwell
+``Liouville's Theorem on Functions with Elementary Integrals''
+Pacific Journal of Mathematics Vol 24 No 1 (1968)
+\verbmsp.org/pjm/1968/241/pjmv24n1p16p.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Ro68.pdf REF:00047
+ abstract = "
+ Defining a function with one variable to be elemetary if it has an
+ explicit representation in terms of a finite number of algebraic
+ operations, logarithms, and exponentials. Liouville's theorem in its
+ simplest case says that if an algebraic function has an elementary
+ integral then the latter is itself an algebraic function plus a sum of
+ constant multiples of logarithms of algebraic functions. Ostrowski has
+ generalized Liouville's results to wider classes of meromorphic
+ functions on regions of the complex plane and J.F. Ritt has given the
+ classical account of the entire subject in his Integraion in Finite
+ Terms, Columbia University Press, 1948. In spite of the essentially
+ algebraic nature of the problem, all proofs so far have been analytic.
+ This paper gives a self contained purely algebraic exposition of the
+ probelm, making a few new points in addition to the resulting
+ simplicity and generalization."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Heck 01]{Hec01} Heck, A.
``Variables in computer algebra, mathematics and science''
The International Journal of Computer Algebra in Mathematics Education
Vol. 8 No. 3 pp195210 (2001)
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Rose72,
+ author = "Rosenlicht, Maxwell",
+ title = "Integration in finite terms",
+ journal = "American Mathematical Monthly",
+ year = "1972",
+ volume = "79",
+ pages = "963972",
+ paper = "Rose72.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Huguet 89]{HP89} Huguet, L.; Poli, A. (eds).
Applied Algebra, Algebraic Algorithms and ErrorCorrecting Codes.
5th International Conference AAECC5 Proceedings.
SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
1989. ISBN 3540510826. LCCN QA268.A35 1987
 keywords = "axiomref",
+\bibitem[Rothstein 76]{Ro76} Rothstein, Michael
+``Aspects of symbolic integration and simplifcation of exponential
+and primitive functions''
+PhD thesis, University of WisconsinMadison (1976)
+\verbwww.cs.kent.edu/~rothstei/dis.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Ro76.pdf REF:00051
+ abstract = "
+ In this thesis we cover some aspects of the theory necessary to obtain
+ a canonical form for functions obtained by integration and
+ exponentiation from the set of rational functions.
\end{chunk}
+ These aspects include a new algorithm for symbolic integration of
+ functions involving logarithms and exponentials which avoids
+ factorization of polynomials in those cases where algebraic extension
+ of the constant field is not required, avoids partial fraction
+ decompositions, and only solves linear systems with a small number of
+ unknowns.
\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ We have also found a theorem which states, roughly speaking, that if
+ integrals which can be represented as logarithms are represented as
+ such, the only algebraic dependence that a new exponential or
+ logarithm can satify is given by the law of exponents or the law of
+ logarithms."
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jacob 93]{JOS93} Jacob, G.; Oussous, N. E.; Steinberg, S. (eds)
Proceedings SC 93
International IMACS Symposium on Symbolic Computation. New Trends and
Developments. LIFL Univ. Lille, Lille France, 1993
 keywords = "axiomref",
+\bibitem[Rothstein 76a]{Ro76a} Rothstein, Michael; Caviness, B.F.
+``A structure theorem for exponential and primitive functions: a preliminary
+ report''
+ACM Sigsam Bulletin Vol 10 Issue 4 (1976)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Ro76a.pdf
+ abstract = "
+ In this paper a generalization of the Risch Structure Theorem is reported.
+ The generalization applies to fields $F(t_1,\ldots,t_n)$ where $F$
+ is a differential field (in our applications $F$ will be a finitely
+ generated extension of $Q$, the field of rational numbers) and each $t_i$
+ is either algebraic over $F_{i1}=F(t_1,\ldots,t_{i1})$, is an
+ exponential of an element in $F_{i1}$, or is an integral of an element
+ in $F_{i1}$. If $t_i$ is an integral and can be expressed using
+ logarithms, it must be so expressed for the generalized structure
+ theorem to apply."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Janssen 88]{Jan88} Jan{\ss}en, R. (ed)
Trends in Computer Algebra, International Symposium
Bad Neuenahr, May 1921, 1987, Proceedings, volume 296 of Lecture Notes in
Computer Science.
SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
1988 ISBN 3540189289, 0387189289 LCCN QA155.7.E4T74 1988
 keywords = "axiomref",
+\bibitem[Rothstein 76b]{Ro76b} Rothstein, Michael; Caviness, B.F.
+``A structure theorem for exponential and primitive functions''
+SIAM J. Computing Vol 8 No 3 (1979)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Ro76b.pdf REF:00104
+ abstract = "
+ In this paper a new theorem is proved that generalizes a result of
+ Risch. The new theorem gives all the possible algebraic relationships
+ among functions that can be built up from the rational functions by
+ algebraic operations, by taking exponentials, and by integration. The
+ functions so generated are called exponential and primitive functions.
+ From the theorem an algorithm for determining algebraic dependence
+ among a given set of exponential and primitive functions is derived.
+ The algorithm is then applied to a problem in computer algebra."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 69]{Jen69} Jenks, R. D.
``META/LISP: An interactive translator writing system''
Research Report International Business Machines, Inc., Thomas J.
Watson Research Center, Yorktown Heights, NY, USA, 1969 RC2968 July 1970
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Roth77,
+ author = "Rothstein, Michael",
+ title = "A new algorithm for the integration of exponential and
+ logarithmic functions",
+ journal = "Proceedings of the 1977 MACSYMA Users Conference",
+ year = "1977",
+ pages = "263274",
+ publisher = "NASA Pub CP2012"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 71]{Jen71} Jenks, R. D.
``META/PLUS: The syntax extension facility for SCRATCHPAD''
Research Report RC 3259, International Business Machines, Inc., Thomas J.
Watson Research Center, Yorktown Heights, NY, USA, 1971
% REF:00040
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Jenks 74]{Jen74} Jenks, R. D.
``The SCRATCHPAD language''
ACM SIGPLAN Notices, 9(4) pp101111 1974 CODEN SINODQ. ISSN 03621340
 keywords = "axiomref",
+\bibitem[Seidenberg 58]{Sei58} Seidenberg, Abraham
+``Abstract differential algebra and the analytic case''
+Proc. Amer. Math. Soc. Vol 9 pp159164 (1958)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jen76]{Jen76} Jenks, Richard D.
``A pattern compiler''
In Richard D. Jenks, editor,
SYMSAC '76: proceedings of the 1976 ACM Symposium on Symbolic and Algebraic
Computation, August 1012, 1976, Yorktown Heights, New York, pp6065,
ACM Press, New York, NY 10036, USA, 1976. LCCN QA155.7.EA .A15 1976
QA9.58.A11 1976
 keywords = "axiomref",
+\bibitem[Seidenberg 69]{Sei69} Seidenberg, Abraham
+``Abstract differential algebra and the analytic case. II''
+Proc. Amer. Math. Soc. Vol 23 pp689691 (1969)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 79]{Jen79} Jenks, R. D.
``MODLISP: An Introduction''
Proc EUROSAM 79, pp466480, 1979 and IBMRC8073 Jan 1980
 keywords = "axiomref",
+\bibitem[Singer 85]{Sing85} Singer, M.F.; Saunders, B.D.; Caviness, B.F.
+``An extension of Liouville's theorem on integration in finite terms''
+SIAM J. of Comp. Vol 14 pp965990 (1985)
+\verbwww4.ncsu.edu/~singer/papers/singer_saunders_caviness.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Sing85.pdf
+ abstract = "
+ In Part 1 of this paper, we give an extension of Liouville's Theorem
+ and give a number of examples which show that integration with special
+ functions involves some phenomena that do not occur in integration
+ with the elementary functions alone. Our main result generalizes
+ Liouville's Theorem by allowing, in addition to the elementary
+ functions, special functions such as the error function, Fresnel
+ integrals and the logarithmic integral (but not the dilogarithm or
+ exponential integral) to appear in the integral of an elementary
+ function. The basic conclusion is that these functions, if they
+ appear, appear linearly. We give an algorithm which decides if an
+ elementary function, built up using only exponential functions and
+ rational operations has an integral which can be expressed in terms of
+ elementary functions and error functions."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 81]{JT81} Jenks, R.D.; Trager, B.M.
``A Language for Computational Algebra''
Proceedings of SYMSAC81, Symposium on Symbolic and Algebraic Manipulation,
Snowbird, Utah August, 1981
 keywords = "axiomref",
+\bibitem[Slagle 61]{Slag61} Slagle, J.
+``A heuristic program that solves symbolic integration problems in
+ freshman calculus''
+Ph.D Diss. MIT, May 1961; also Computers and Thought, Feigenbaum and Feldman.
+% REF:00014
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 81a]{JT81a} Jenks, R.D.; Trager, B.M.
``A Language for Computational Algebra''
SIGPLAN Notices, New York: Association for Computing Machiner, Nov 1981
 keywords = "axiomref",
+\bibitem[Terelius 09]{Tere09} Terelius, Bjorn
+``Symbolic Integration''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Tere09.pdf
+ abstract = "
+ Symbolic integration is the problem of expressing an indefinite integral
+ $\int{f}$ of a given function $f$ as a finite combination $g$ of elementary
+ functions, or more generally, to determine whether a certain class of
+ functions contains an element $g$ such that $g^\prime = f$.
+
+ In the first part of this thesis, we compare different algorithms for
+ symbolic integration. Specifically, we review the integration rules
+ taught in calculus courses and how they can be used systematically to
+ create a reasonable, but somewhat limited, integration method. Then we
+ present the differential algebra required to prove the transcendental
+ cases of Risch's algorithm. Risch's algorithm decides if the integral
+ of an elementary function is elementary and if so computes it. The
+ presentation is mostly selfcontained and, we hope, simpler than
+ previous descriptions of the algorithm. Finally, we describe
+ RischNorman's algorithm which, although it is not a decision
+ procedure, works well in practice and is considerably simpler than the
+ full Risch algorithm.
+
+ In the second part of this thesis, we briefly discuss an
+ implementation of a computer algebra system and some of the
+ experiences it has given us. We also demonstrate an implementation of
+ the rulebased approach and how it can be used, not only to compute
+ integrals, but also to generate readable derivations of the results."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 81b]{JT81b} Jenks, R.D.; Trager, B.M.
``A Language for Computational Algebra''
IBM Research Report RC8930 IBM Yorktown Heights, NY
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Trag76,
+ author = "Trager, Barry",
+ title = "Algebraic factoring and rational function integration",
+ journal = "Proceedings of SYMSAC'76",
+ year = "1976",
+ pages = "219226",
+ paper = "Trag76.pdf",
+ abstract = "
+ This paper presents a new, simple, and efficient algorithm for
+ factoring polynomials in several variables over an algebraic number
+ field. The algorithm is then used interatively to construct the
+ splitting field of a polynomial over the integers. Finally the
+ factorization and splitting field algorithms are applied to the
+ problem of determining the transcendental part of the integral of a
+ rational function. In particular, a constructive procedure is given
+ for finding a least degree extension field in which the integral can
+ be expressed."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 84a]{Jen84a} Jenks, Richard D.
``The new SCRATCHPAD language and system for computer algebra''
In Golden and Hussain [GH84], pp409??
 keywords = "axiomref",
+\bibitem[Trager 76a]{Tr76a} Trager, Barry Marshall
+``Algorithms for Manipulating Algebraic Functions''
+MIT Master's Thesis.
+\verbwww.dm.unipi.it/pages/gianni/public_html/AlgComp/fattorizzazioneEA.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Tr76a.pdf REF:00050
+ abstract = "
+ Given a base field $k$, of characteristic zero, with effective
+ procedures for performing arithmetic and factoring polynomials, this
+ thesis presents algorithms for extending those capabilities to
+ elements of a finite algebraic symbolic manipulation system. An
+ algebraic factorization algorithm along with a constructive version of
+ the primitive element theorem is used to construct splitting fields of
+ polynomials. These fields provide a context in which we can operate
+ symbolically with all the roots of a set of polynomials. One
+ application for this capability is rational function integrations.
+ Previously presented symbolic algorithms concentrated on finding the
+ rational part and were only able to compute the complete
+ integral in special cases. This thesis presents an algorithm for
+ finding an algebraic extension field of least degreee in which the
+ integral can be expressed, and then constructs the integral in that
+ field. The problem of algebraic function integration is also
+ examined, and a highly efficient procedure is presented for generating
+ the algebraic part of integrals whose function fields are defined by a
+ single radical extension of the rational functions."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 84b]{Jen84b} Jenks, Richard D.
``A primer: 11 keys to New Scratchpad''
In Fitch [Fit84], pp123147. ISBN 038713350X LCCN QA155.7.E4 I57 1984
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@phdthesis{Trag84,
+ author = "Trager, Barry",
+ title = "On the integration of algebraic functions",
+ school = "MIT",
+ year = "1984",
+ url = "http://www.dm.unipi.it/pages/gianni/public_html/AlgComp/thesis.pdf",
+ paper = "Trag76.pdf",
+ abstract = "
+ We show how the ``rational'' approach for integrating algebraic
+ functions can be extended to handle elementary functions. The
+ resulting algorithm is a practical decision procedure for determining
+ whether a given elementary function has an elementary antiderivative,
+ and for computing it if it exists."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 86]{JWS86} Jenks, Richard D.; Sutor, Robert S.;
Watt, Stephen M.
``Scratchpad II: An Abstract Datatype System for Mathematical Computation''
Research Report RC 12327 (\#55257), Iinternational Business Machines, Inc.,
Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1986 23pp
\verbwww.csd.uwo.ca/~watt/pub/reprints/1987imaspadadt.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/JWS86.pdf
 keywords = "axiomref",
+\bibitem[W\"urfl 07]{Wurf07} W\"urfl, Andreas
+``Basic Concepts of Differential Algebra''
+\verbwww14.in.tum.de/konferenzen/Jass07/courses/1/Wuerfl/wuerfl_paper.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Wurf07.pdf
+ abstract = "
+ Modern computer algebra systems symbolically integrate a vast variety
+ of functions. To reveal the underlying structure it is necessary to
+ understand infinite integration not only as an analytical problem but
+ as an algebraic one. Introducing the differential field of elementary
+ functions we sketch the mathematical tools like Liouville's Principle
+ used in modern algorithms. We present Hermite's method for integration
+ of rational functions as well as the Rothstein/Trager method for
+ rational and for elementary functions. Further applications of the
+ mentioned algorithms in the field of ODE's conclude this paper."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Scratchpad II is an abstract datatype language and system that is
under development in the Computer Algebra Group, Mathematical Sciences
Department, at the IBM Thomas J. Watson Research Center. Some features
of APL that made computation particularly elegant have been borrowed.
Many different kinds of computational objects and data structures are
provided. Facilities for computation include symbolic integration,
differentiation, factorization, solution of equations and linear
algebra. Code economy and modularity is achieved by having
polymorphic packages of functions that may create datatypes. The use
of categories makes these facilities as general as possible.
\end{adjustwidth}
+\subsection{Partial Fraction Decomposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Jenks 87]{JWS87} Jenks, Richard D.; Sutor, Robert S.;
Watt, Stephen M.
``Scratchpad II: an Abstract Datatype System for Mathematical Computation''
Proceedings Trends in Computer Algebra, Bad Neuenahr, LNCS 296,
Springer Verlag, (1987)
 keywords = "axiomref",
+\bibitem[Angell]{Angell} Angell, Tom
+``Guidelines for Partial Fraction Decomposition''
+\verbwww.math.udel.edu/~angell/partfrac_I.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Angell.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 88]{JSW88} Jenks, R. D.; Sutor, R. S.; Watt, S. M.
``Scratchpad II: An abstract datatype system for mathematical computation''
In Jan{\ss}en [Jan88],
pp12?? ISBN 3540189289, 0387189289 LCCN QA155.7.E4T74 1988
 keywords = "axiomref",
+\bibitem[Laval 08]{Lava08} Laval, Philippe B.
+``Partial Fractions Decomposition''
+\verbwww.math.wisc.edu/~park/Fall2011/integration/Partial%20Fraction.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Lava08.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 88a]{Jen88a} Jenks, R. D.
``A Guide to Programming in BOOT''
Computer Algebra Group, Mathematical Sciences Department, IBM Research
Draft September 5, 1988
 keywords = "axiomref",
+\bibitem[Mudd 14]{Mudd14} Harvey Mudd College
+``Partial Fractions''
+\verbwww.math.hmc.edu/calculus/tutorials/partial_fractions/partial_fractions.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Mudd14.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 88b]{Jen88b} Jenks, Richard
``The Scratchpad II Computer Algebra System Interactive Environment Users
Guide''
 Spring 1988
 keywords = "axiomref",
+\bibitem[Rajasekaran 14]{Raja14} Rajasekaran, Raja
+``Partial Fraction Expansion''
+\verbwww.utdallas.edu/~raja1/EE4361%20Spring%2014/Lecture%20Notes/
+\verbPartial%20Fractions.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Raja14.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 88c]{JWS88} Jenks, R. D.; Sutor, R. S.; Watt, S. M.
``Scratchpad II: an abstract datatype system for mathematical computation''
In Jan{\ss}en
[Jan88], pp1237. ISBN 3540189289, 0387189289 LCCN QA155.7.E4T74 1988
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{axiom.bib}
@book{Jenk92,
 author = "Jenks, Richard D. and Sutor, Robert S.",
 title = "AXIOM: The Scientific Computation System",
 publisher = "SpringerVerlag, Berlin, Germany",
 year = "1992",
 isbn = "0387978550",
 keywords = "axiomref"
}
+\bibitem[Wootton 14]{Woot14} Wootton, Aaron
+``Integration of Rational Functions by Partial Fractions''
+\verbfaculty.up.edu/wootton/calc2/section7.4.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Woot14.pdf
\end{chunk}
+\subsection{Ore Rings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Jenks 94]{JT94} Jenks, R. D.; Trager, B. M.
``How to make AXIOM into a Scratchpad''
In ACM [ACM94], pp3240 ISBN 0897916387 LCCN QA76.95.I59 1994
%\verbaxiomdeveloper.org/axiomwebsite/papers/JT94.pdf
 keywords = "axiomref",

\end{chunk}
+This is used as a reference for the LeftOreRing category, in particular,
+the least left common multiple (lcmCoef) function.
\begin{chunk}{ignore}
\bibitem[Joswig 03]{JT03} Joswig, Michael; Takayama, Nobuki
``Algebra, geometry, and software systems''
SpringerVerlag ISBN 3540002561 p291
 keywords = "axiomref",
+\bibitem[Abramov 97]{Abra97} Abramov, Sergei A.; van Hoeij, Mark
+``A method for the Integration of Solutions of Ore Equations''
+Proc ISSAC 97 pp172175 (1997)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Abra97.pdf
+ abstract = "
+ We introduce the notion of the adjoint Ore ring and give a definition
+ of adjoint polynomial, operator and equation. We apply this for
+ integrating solutions of Ore equations."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Joyner 06]{J006} Joyner, David
``OSCAS  Maxima''
SIGSAM Communications in Computer Algebra, 157 2006
\verbsage.math.washington.edu/home/wdj/sigsam/oscascca1.pdf
 keywords = "axiomref",
+\bibitem[Delenclos 06]{DL06} Delenclos, Jonathon; Leroy, Andr\'e
+``Noncommutative Symmetric functions and $W$polynomials''
+\verbarxiv.org/pdf/math/0606614.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/DL06.pdf
+ abstract = "
+ Let $K$, $S$, $D$ be a division ring an endomorphism and a
+ $S$derivation of $K$, respectively. In this setting we introduce
+ generalized noncommutative symmetric functions and obtain Vi\'ete
+ formula and decompositions of different operators. $W$polynomials
+ show up naturally, their connetions with $P$independency. Vandermonde
+ and Wronskian matrices are briefly studied. The different linear
+ factorizations of $W$polynomials are analysed. Connections between
+ the existence of LLCM (least left common multiples) of monic linear
+ polynomials with coefficients in a ring and the left duo property are
+ established at the end of the paper."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Joyner 14]{JO14} Joyner, David
``Links to some open source mathematical programs''
\verbwww.opensourcemath.org/opensource_math.html
 keywords = "axiomref",
+\bibitem[Abramov 05]{Abra05} Abramov, S.A.; Le, H.Q.; Li, Z.
+``Univariate Ore Polynomial Rings in Computer Algebra''
+\verbwww.mmrc.iss.ac.cn/~zmli/papers/oretools.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Abra05.pdf
+ abstract = "
+ We present some algorithms related to rings of Ore polynomials (or,
+ briefly, Ore rings) and describe a computer algebra library for basic
+ operations in an arbitrary Ore ring. The library can be used as a
+ basis for various algorithms in Ore rings, in particular, in
+ differential, shift, and $q$shift rings."
\end{chunk}
\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Number Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Kauers 08]{Kau08} Kauers, Manuel
``Integration of Algebraic Functions: A Simple Heuristic for Finding
the Logarithmic Part''
ISSAC July 2008 ACM 978159593904 pp133140
\verbwww.risc.jku.at/publications/download/risc_3427/Ka01.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Kau08.pdf
 keywords = "axiomref",
+\bibitem[Shoup 08]{Sho08} Shoup, Victor
+``A Computational Introduction to Number Theory''
+\verbshoup.net/ntb/ntbv2.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Sho08.pdf
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A new method is proposed for finding the logarithmic part of an
integral over an algebraic function. The method uses Gr\"obner bases
and is easy to implement. It does not have the feature of finding a
closed form of an integral whenever there is one. But it very often
does, as we will show by a comparison with the builtin integrators of
some computer algebra systems.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Keady 94]{KN94} Keady, G.; Nolan, G.
``Production of Argument SubPrograms in the AXIOM  NAG
link: examples involving nonleanr systems''
Technical Report TR1/94
ATR/7 (NP2680), Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and
Oxford, UK, 1994
\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
 keywords = "axiomref",
+\subsection{Polynomial Factorization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{chunk}
+\subsection{Branch Cuts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Kelsey 99]{Kel99} Kelsey, Tom
``Formal Methods and Computer Algebra: A Larch Specification of AXIOM
Categories and Functors''
Ph.D. Thesis, University of St Andrews, 1999
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Beau03,
+ author = "Beaumont, James and Bradford, Russell and Davenport, James H.",
+ title = "Better simplification of elementary functions through power series",
+ journal = "2003 International Symposium on Symbolic and Algebraic Computation",
+ series = "ISSAC'03",
+ year = "2003",
+ month = "August",
+ paper = "Beau03.pdf",
+ abstract = "
+ In [5], we introduced an algorithm for deciding whether a proposed
+ simplification of elementary functions was correct in the presence of
+ branch cuts. This algorithm used multivalued function simplification
+ followed by verification that the branches were consistent.
\end{chunk}
+ In [14] an algorithm was presented for zerotesting functions defined
+ by ordinary differential equations, in terms of their power series.
\begin{chunk}{ignore}
\bibitem[Kelsey 00a]{Kel00a} Kelsey, Tom
``Formal specification of computer algebra''
University of St Andrews, 6th April 2000
\verbwww.cs.standrews.cs.uk/~tom/pub/fscbs.ps
%\verbaxiomdeveloper.org/axiomwebsite/papers/Kel00a.pdf
 keywords = "axiomref",
+ The purpose of the current paper is to investigate merging the two
+ techniques. In particular, we will show an explicit reduction to the
+ constant problem [16]."
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We investigate the use of formal methods languages and tools in the
design and development of computer algebra systems (henceforth CAS).
We demonstrate that errors in CAS design can be identified and
corrected by the use of (i) abstract specifications of types and
procedures, (ii) automated proofs of properties of the specifications,
and (iii) interface specifications which assist the verification of
pre and post conditions of implemented code.
\end{adjustwidth}
+\begin{chunk}{axiom.bib}
+@article{Beau07,
+ author = "Beaumont, James C. and Bradford, Russell J. and
+ Davenport, James H. and Phisanbut, Nalina",
+ title = "Testing elementary function identities using CAD",
+ journal = "Applicable Algebra in Engineering, Communication and Computing",
+ year = "2007",
+ volume = "18",
+ number = "6",
+ issn = "09381279",
+ publisher = "SpringerVerlag",
+ pages = "513543",
+ paper = "Beau07.pdf",
+ abstract = "
+ One of the problems with manipulating function identities in computer
+ algebra systems is that they often involve functions which are
+ multivalued, whilst most users tend to work with singlevalued
+ functions. The problem is that many wellknown identities may no
+ longer be true everywhere in the complex plane when working with their
+ singlevalued counterparts. Conversely, we cannot ignore them, since
+ in particular contexts they may be valid. We investigate the
+ practicality of a method to verify such identities by means of an
+ experiment; this is based on a set of test examples which one might
+ realistically meet in practice. Essentially, the method works as
+ follows. We decompose the complex plane via means of cylindrical
+ algebraic decomposition into regions with respect to the branch cuts
+ of the functions. We then test the identity numerically at a sample
+ point in the region. The latter step is facilitated by the notion of
+ the {\sl adherence} of a branch cut, which was previously introduced
+ by the authors. In addition to presenting the results of the
+ experiment, we explain how adherence relates to the proposal of
+ {\sl signed zeros} by W. Kahan, and develop this idea further in order to
+ allow us to cover previously untreatable cases. Finally, we discuss
+ other ways to improve upon our general methodology as well as topics
+ for future research."
+}
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Kelsey 00b]{Kel00b} Kelsey, Tom
``Formal specification of computer algebra''
(slides) University of St Andrews, Sept 21, 2000
\verbwww.cs.standrews.cs.uk/~tom/pub/fscbstalk.ps
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Kendall 99a]{Ken99a} Kendall, W.S.
``Itovsn3 in AXIOM: modules, algebras and stochastic differentials''
\verbwww2.warwick.ac.uk/fac/sci/statistics/staff/academicresearch/
\verbkendall/personal/ppt/328.ps.gz
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Kendall 99b]{Ken99b} Kendall, W.S.
``Symbolic It\^o calculus in AXIOM: an ongoing story
\verbwww2.warwick.ac.uk/fac/sci/statistics/staff/academicresearch/
\verbkendall/personal/ppt/327.ps.gz
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Kosleff 91]{Kos91} P.V. Koseleff
``Word games in free Lie algebras: several bases and formulas''
Theoretical Computer Science 79(1) pp241256 Feb. 1991 CODEN TCSCDI
ISSN 03043975
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Kusche 89]{KKM89} Kusche, K.; Kutzler, B.; Mayr, H.
``Implementation of a geometry theorem proving package in SCRATCHPAD II''
In Davenport [Dav89] pp246257 ISBN 3540515178 LCCN QA155.7.E4E86 1987
 keywords = "axiomref",

\end{chunk}

\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Lahey 08]{Lah08} Lahey, Tim
``Sage Integration Testing''
\verbgithub.com/tjl/sage_int_testing Dec. 2008
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Lambe 89]{Lam89} Lambe, L. A.
``Scratchpad II as a tool for mathematical research''
Notices of the AMS, February 1928 pp143147
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Lambe 91]{Lam91} Lambe, L. A.
``Resolutions via homological perturbation''
Journal of Symbolic Computation 12(1) pp7187 July 1991
CODEN JSYCEH ISSN 07477171
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Lambe 92]{Lam92} Lambe, Larry
``Next Generation Computer Algebra Systems AXIOM and the Scratchpad
Concept: Applications to Research in Algebra''
$21^{st}$ Nordic Congress of Mathematicians 1992
%\verbaxiomdeveloper.org/axiomwebsite/papers/Lam92.pdf
 keywords = "axiomref",

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
One way in which mathematicians deal with infinite amounts of data is
symbolic representation. A simple example is the quadratic equation
\[x = \frac{b\pm\sqrt{b^24ac}}{2a}\]
a formula which uses symbolic representation to describe the solutions
to an infinite class of equations. Most computer algebra systems can
deal with polynomials with symbolic coefficients, but what if symbolic
exponents are called for (e.g. $1+t^i$)? What if symbolic limits on
summations are also called for, for example
\[1+t+\ldots+t^i=\sum_j{t^j}\]
The ``Scratchpad Concept'' is a theoretical ideal which allows the
implementation of objects at this level of abstraction and beyond in a
mathematically consistent way. The Axiom computer algebra system is an
implementation of a major part of the Scratchpad Concept. Axiom
(formerly called Scratchpad) is a language with extensible
parameterized types and generic operators which is based on the
notions of domains and categories. By examining some aspects of the
Axiom system, the Scratchpad Concept will be illustrated. It will be
shown how some complex problems in homologicial algebra were solved
through the use of this system.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Lambe 93]{Lam93} Lambe, Larry
``On Using Axiom to Generate Code''
(preprint) 1993
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Lambe 93a]{LL93} Lambe, Larry; Luczak, Richard
``ObjectOriented Mathematical Programming and Symbolic/Numeric Interface''
$3^{rd}$ International Conf. on Expert Systems in Numerical Computing 1993
%\verbaxiomdeveloper.org/axiomwebsite/papers/LL93.pdf
 keywords = "axiomref",

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Lebedev 08]{Leb08} Lebedev, Yuri
``OpenMath Library for Computing on Riemann Surfaces''
PhD thesis, Nov 2008 Florida State University
\verbwww.math.fsu.edu/~ylebedev/research/HyperbolicGeometry.html
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[LeBlanc 91]{LeB91} LeBlanc, S.E.
``The use of MathCAD and Theorist in the ChE classroom''
In Anonymous [Ano91], pp287299 (vol. 1) 2 vols.
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Lecerf 96]{Le96} Lecerf, Gr\'egoire
``Dynamic Evaluation and Real Closure Implementation in Axiom''
June 29, 1996
\verblecerf.perso.math.cnrs.fr/software/drc/drc.ps
%\verbaxiomdeveloper.org/axiomwebsite/papers/Le96.ps
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Lecerf 96a]{Le96a} Lecerf, Gr\'egoire
``The Dynamic Real Closure implemented in Axiom''
\verblecerf.perso.math.cnrs.fr/software/drc/drc.ps
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Levelt 95]{Lev95} Levelt, A. H. M. (ed)
ISSAC '95: Proceedings of the 1995 International
Symposium on Symbolic and Algebraic Computation: July 1012, 1995, Montreal,
Canada ISSACPROCEEDINGS1995. ACM Press, New York, NY 10036, USA, 1995
ISBN 0897916999 LCCN QA76.95 I59 1995 ACM order number 505950
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Li 06]{LM06} Li, Xin; Maza, Moreno
``Efficient Implementation of Polynomial Arithmetic in a MultipleLevel
Programming Environment''
Lecture Notes in
Computer Science Springer Vol 4151/2006 ISBN 9783540380849 pp1223
Proceedings of International Congress of Mathematical Software ICMS 2006
\verbwww.csd.uwo.ca/~moreno//Publications/LiMorenoMazaICMS06.pdf
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Li 10]{YL10} Li, Yue; Dos Reis, Gabriel
``A Quantitative Study of Reductions in Algebraic Libraries''
PASCO 2010
\verbwww.axiomatics.org/~gdr/concurrency/quantpasco10.pdf
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Li 11]{YL11} Li, Yue; Dos Reis, Gabriel
``An Automatic Parallelization Framework for Algebraic Computation
Systems''
ISSAC 2011
\verbwww.axiomatics.org/~gdr/concurrency/oaconcissac11.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/YL11.pdf
 keywords = "axiomref",

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
This paper proposes a nonintrusive automatic parallelization
framework for typeful and propertyaware computer algebra systems.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Ligatsikas 96]{Liga96} Ligatsikas, Zenon; Rioboo, Renaud;
Roy, Marie Francoise
``Generic computation of the real closure of an ordered field''
Math. and Computers in Simulation 42 pp 541549 (1996)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Liga96.pdf
 keywords = "axiomref",

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
This paper describes a generalization of the real closure computation
of an ordered field (Rioboo, 1991) enabling to use different technques
to code a single real algebraic number.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Linton 93]{Lin93} Linton, Steve
``Vector Enumeration Programs, version 3.04''
\verbwww.cs.standrews.ac.uk/~sal/nme/nme_toc.html#SEC1
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Liska 97]{LD97} Liska, Richard; Drska, Ladislav; Limpouch, Jiri;
Sinor, Milan; Wester, Michael; Winkler, Franz
``Computer Algebra  algorithms, systems and applications''
June 2, 1997
\verbkfe.fjfi.cvut.cz/~liska/ca/all.html
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Lucks 86]{Luc86} Lucks, Michael
``A fast implementation of polynomial factorization''
In Bruce W. Char, editor, Proceedings of the 1986 Symposium on Symbolic
and Algebraic Computation: SYMSAC '86, July 2123, 1986, Waterloo, Ontario,
pp228232 ACM Press, New York, NY 10036, USA, 1986. ISBN 0897911997
LCCN QA155.7.E4 A281 1986 ACM order number 505860
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Lueken 77]{Lue77} Lueken, E.
``Ueberlegungen zur Implementierung eines Formelmanipulationssystems''
Master's thesis, Technischen Universit{\"{a}}t CaroloWilhelmina zu
Braunschweig. Braunschweig, Germany, 1977
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Lynch 91]{LM91} Lynch, R.; Mavromatis, H. A.
``New quantum mechanical perturbation technique
using an 'electronic scratchpad' on an inexpensive computer''
American Journal of Pyhsics, 59(3) pp270273, March 1991.
CODEN AJPIAS ISSN 00029505
 keywords = "axiomref",

\end{chunk}

\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Mahboubi 05]{Mah05} Mahboubi, Assia
``Programming and certifying the CAD algorithm inside the coq system''
Mathematics, Algorithms, Proofs, volume 05021 of Dagstuhl
Seminar Proceedings, Schloss Dagstuhl (2005)
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Mathews 89]{Mat89} Mathews, J.
``Symbolic computational algebra applied to Picard iteration''
Mathematics and computer education, 23(2) pp117122 Spring 1989 CODEN MCEDDA,
ISSN 07308639
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[McJones 11]{McJ11} McJones, Paul
``Software Presentation Group  Common Lisp family''
\verbwww.softwarepreservation.org/projects/LISP/common_lisp_family
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Melachrinoudis 90]{MR90} Melachrinoudis, E.; Rumpf, D. L.
``Teaching advantages of transparent computer software  MathCAD''
CoED, 10(1) pp7176, JanuaryMarch 1990 CODEN CWLJDP ISSN 07368607
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Miola 90]{Mio90} Miola, A. (ed)
``Design and Implementation of Symbolic Computation Systems''
International Symposium DISCO '90, Capri, Italy, April 1012, 1990, Proceedings
volume 429 of Lecture Notes in Cmputer Science,
SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
1990 ISBN 0387525319 (New York), 3540525319 (Berlin) LCCN QA76.9.S88I576
1990
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Miola 93]{Mio93} Miola, A. (ed)
``Design and Implementation of Symbolic Computation Systems''
International Symposium DISCO '93 Gmunden, Austria, September 1517, 1993:
Proceedings.
SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
1993 ISBN 354057235X LCCN QA76.9.S88I576 1993
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Missura 94]{Miss94} Missura, Stephan A.; Weber, Andreas
``Using Commutativity Properties for Controlling Coercions''
\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/
\verbWeberA/MissuraWeber94a.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Miss94.pdf
 keywords = "axiomref",

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
This paper investigates some soundness conditions which have to be
fulfilled in systems with coercions and generic operators. A result of
Reynolds on unrestricted generic operators is extended to generic
operators which obey certain constraints. We get natural conditions
for such operators, which are expressed within the theoretic framework
of category theory. However, in the context of computer algebra, there
arise examples of coercions and generic operators which do not fulfil
these conditions. We describe a framework  relaxing the above
conditions  that allows distinguishing between cases of ambiguities
which can be resolved in a quite natural sense and those which
cannot. An algorithm is presented that detects such unresolvable
ambiguities in expressions.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Monagan 87]{Mon87} Monagan, Michael B.
``Support for Data Structures in Scratchpad II''
in [Wit87], pp1718
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Monagan 93]{Mon93} Monagan, M. B.
``Gauss: a parameterized domain of computation system with
support for signature functions''
In Miola [Mio93], pp8194 ISBN 354057235X LCCN QA76.9.S88I576 1993
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Mora 89]{Mor89} Mora, T. (ed)
Applied Algebra, Algebraic Algorithms and ErrorCorrecting
Codes, 6th International Conference, AAECC6, Rome, Italy, July 48, 1998,
Proceedings, volume 357 of Lecture Notes in Computer Science
SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
1989 ISBN 3540510834, LCCN QA268.A35 1988 Conference held jointly with
ISSAC '88
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Moses 71]{Mos71} Moses, Joel
``Algebraic Simplification: A Guide for the Perplexed''
CACM August 1971 Vol 14 No. 8 pp527537
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Moses 08]{Mos08} Moses, Joel
``Macsyma: A Personal History''
Invited Presentation in Milestones in Computer Algebra, May 2008, Tobago
\verbesd.mit.edu/Faculty_Pages/moses/Macsyma.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Mos08.pdf
 keywords = "axiomref",

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
The Macsyma system arose out of research on mathematical software in
the AI group at MIT in the 1960's. Algorithm development in symbolic
integration and simplification arose out of the interest of people,
such as the author, who were also mathematics students. The later
development of algorithms for the GCD of sparse polynomials, for
example, arose out of the needs of our user community. During various
times in the 1970's the computer on which Macsyma ran was one of the
most popular notes on the ARPANET. We discuss the attempts in the late
70's and the 80's to develop Macsyma systems that ran on popular
computer architectures. Finally, we discuss the impact of the
fundamental ideas in Macsyma on current research on large scale
engineering systems.
\end{adjustwidth}

\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Naylor]{NPxx} Naylor, William; Padget, Julian
``From Untyped to Polymorphically Typed Objects in Mathematical Web
Services''
%\verbaxiomdeveloper.org/axiomwebsite/papers/NPxx.pdf
 keywords = "axiomref",

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Naylor 95]{N95} Naylor, Bill
``Symbolic Interface for an advanced hyperbolic PDE solver''
\verbwww.sci.csd.uwo.ca/~bill/Papers/symbInterface2.ps
%\verbaxiomdeveloper.org/axiomwebsite/papers/N95.pdf
 keywords = "axiomref",

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
An Axiom front end is described, which is used to generate
mathematical objects needed by one of the latest NAG routines, to be
included in the Mark 17 version of the NAG Numerical library. This
routine uses powerful techniques to find the solution to Hyperbolic
Partial Differential Equations in conservation form and in one spatial
dimension. These mathematical objects are nontrivial, requiring much
mathematical knowledge on the part of the user, which is otherwise
irrelvant to the physical problem which is to be solved. We discuss
the individual mathematical objects, considering the mathematical
theory which is relevant, and some of the problems which have been
encountered and solved during the FORTRAN generation necessary to
realise the object. Finally we display some of our results.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Naylor 00b]{ND00} Naylor, W.A.; Davenport, J.H.
``A MonteCarlo Extension to a CategoryBased Type System''
\verbwww.sci.csd.uwo.ca/~bill/Papers/monteCarCat3.ps
%\verbaxiomdeveloper.org/axiomwebsite/papers/ND00.pdf
 keywords = "axiomref",

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
The normal claim for mathematics is that all calculations are 100\%
accurate and therefore one calculation can rely completely on the
results of subcalculations, hoever there exist {\sl MonteCarlo}
algorithms which are often much faster than the equivalent
deterministic ones where the results will have a prescribed
probability (presumably small) of being incorrect. However there has
been little discussion of how such algorithms can be used as building
blocks in Computer Algebra. In this paper we describe how the
computational category theory which is the basis of the type structure
used in the Axiom computer algebra system may be extended to cover
probabilistic algorithms, which use MonteCarlo techniques. We follow
this with a specific example which uses Straight Line Program
representation.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Norman 75]{Nor75} Norman, A. C.
``Computing with formal power series''
ACM Transactions on Mathematical Software, 1(4) pp346356
Dec. 1975 CODEN ACMSCU ISSN 00983500
 keywords = "axiomref",

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Norman 75a]{Nor75a} Norman, A.C.
``The SCRATCHPAD Power Series Package''
IBM T.J. Watson Research RC4998
 keywords = "axiomref",

\end{chunk}

\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Ollivier 89]{Oll89} Ollivier, F.
``Inversibility of rational mappings and structural
identifiablility in automatics''
In ACM [ACM89], pp4354 ISBN 0897913256 LCCN QA76.95.I59 1989
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Brad02,
+ author="Bradford, Russell and Corless, RobertM. and Davenport, JamesH. and
+ Jeffrey, DavidJ. and Watt, StephenM.",
+ title="Reasoning about the Elementary Functions of Complex Analysis",
+ journal="Annals of Mathematics and Artificial Intelligence",
+ year="2002",
+ issn="10122443",
+ volume="36",
+ number="3",
+ doi="10.1023/A:1016007415899",
+ url="http://dx.doi.org/10.1023/A%3A1016007415899",
+ publisher="Kluwer Academic Publishers",
+ keywords="elementary functions; branch cuts; complex identities",
+ pages="303318",
+ paper = "Brad02.pdf",
+ abstract = "
+ There are many problems with the simplification of elementary
+ functions, particularly over the complex plane, though not
+ exclusively. 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 complex
+ variable analysis."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Online 72]{Onl72}.
Online 72: conference proceedings ... international conference on online
interactive computing, Brunel University, Uxbridge, England, 47 September
1972 ISBN 0903796023 LCCN QA76.55.O54 1972 Two volumes.
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@inproceedings{Chyz11,
+ author = "Chyzak, Fr\'ed\'eric and Davenport, James H. and Koutschan, Christoph and Salvy, Bruno",
+ title = "On Kahan's Rules for Determining Branch Cuts",
+ booktitle = "Proc. 13th Int. Symp. on Symbolic and Numeric Algorithms for Scientific Computing",
+ year = "2011",
+ isbn = "9781467302074",
+ location = "Timisoara",
+ pages = "4751",
+ doi = "10.1109/SYNASC.2011.51",
+ acmid = "258794",
+ publisher = "IEEE",
+ paper = "Chyz11.pdf",
+ abstract = "
+ In computer algebra there are different ways of approaching the
+ mathematical concept of functions, one of which is by defining them as
+ solutions of differential equations. We compare different such
+ appraoches and discuss the occurring problems. The main focus is on
+ the question of determining possible branch cuts. We explore the
+ extent to which the treatment of branch cuts can be rendered (more)
+ algorithmic, by adapting Kahan's rules to the differential equation
+ setting."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[OpenMath]{OpenMa}.
``OpenMath Technical Overview''
\verbwww.openmath.org/overview/technical.html
 keywords = "axiomref",

+\begin{chunk}{axiom.bib}
+@article{Dave10,
+ author = "Davenport, James",
+ title = {The Challenges of Multivalued "Functions"},
+ journal = "Lecture Notes in Computer Science",
+ volume = "6167",
+ year = "2010",
+ pages = "112",
+ paper = "Dave10.pdf",
+ abstract = "
+ Although, formally, mathematics is clear that a function is a
+ singlevalued object, mathematical practice is looser, particularly
+ with nth roots and various inverse functions. In this paper, we point
+ out some of the looseness, and ask what the implications are, both for
+ Artificial Intelligence and Symbolic Computation, of these practices.
+ In doing so, we look at the steps necessary to convert existing tests
+ into
+ \begin{itemize}
+ \item (a) rigorous statements
+ \item (b) rigorously proved statements
+ \end{itemize}
+ In particular we ask whether there might be a constant ``de Bruij factor''
+ [18] as we make these texts more formal, and conclude that the answer
+ depends greatly on the interpretation being placed on the symbols."
+}
+
\end{chunk}
\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{axiom.bib}
+@article{Dave12,
+ author = "Davenport, James H. and Bradford, Russell and England, Matthew
+ and Wilson, David",
+ title = "Program Verification in the presence of complex numbers, functions
+ with branch cuts etc",
+ journal = "14th Int. Symp. on Symbolic and Numeric Algorithms for
+ Scientific Computing",
+ year = "2012",
+ series = "SYNASC'12",
+ pages = "8388",
+ publisher = "IEEE",
+ paper = "Dave12.pdf",
+ abstract = "
+ In considering the reliability of numerical programs, it is normal to
+ ``limit our study to the semantics dealing with numerical precision''.
+ On the other hand, there is a great deal of work on the reliability of
+ programs that essentially ignores the numerics. The thesis of this
+ paper is that there is a class of problems that fall between the two,
+ which could be described as ``does the lowlevel arithmetic implement
+ the highlevel mathematics''. Many of these problems arise because
+ mathematics, particularly the mathematics of the complex numbers, is
+ more difficult than expected; for example the complex function log is
+ not continuous, writing down a program to compute an inverse function
+ is more complicated than just solving an equation, and many algebraic
+ simplification rules are not universally valid.
\begin{chunk}{ignore}
\bibitem[Page 07]{Pa07} Page, William S.
``Axiom  Open Source Computer Algebra System''
Poster ISSAC 2007 Proceedings Vol 41 No 3 Sept 2007 p114
 keywords = "axiomref",
+ The good news is that these problems are theoretically capable of
+ being solved, and are practically close to being solved, but not yet
+ solved, in several realworld examples. However, there is still a long
+ way to go before implementations match the theoretical possibilities."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Petitot 90]{Pet90} Petitot, Michel
``Types r\'ecursifs en scratchpad, application aux polyn\^omes non
commutatifs''
LIFL, 1990
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Jeff04,
+ author = "Jeffrey, D. J. and Norman, A. C.",
+ title = "Not Seeing the Roots for the Branches: Multivalued Functions in
+ Computer Algebra",
+ journal = "SIGSAM Bull.",
+ issue_date = "September 2004",
+ volume = "38",
+ number = "3",
+ month = "September",
+ year = "2004",
+ issn = "01635824",
+ pages = "5766",
+ numpages = "10",
+ url = "http://doi.acm.org/10.1145/1040034.1040036",
+ doi = "10.1145/1040034.1040036",
+ acmid = "1040036",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ paper = "Jeff04.pdf",
+ abstract = "
+ We discuss the multiple definitions of multivalued functions and their
+ suitability for computer algebra systems. We focus the discussion by
+ taking one specific problem and considering how it is solved using
+ different definitions. Our example problem is the classical one of
+ calculating the roots of a cubic polynomial from the Cardano formulae,
+ which contains fractional powers. We show that some definitions of
+ these functions result in formulae that are correct only in the sense
+ that they give candidates for solutions; these candidates must then be
+ tested. Formulae that are based on singlevalued functions, in
+ contract, are efficient and direct."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Petitot 93]{Pet93} Petitot, M.
``Experience with Axiom''
In Jacob et al. [JOS93], page 240
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@inproceedings{Kaha86,
+ author = "Kahan, W.",
+ title = "Branch cuts for complex elementary functions",
+ booktitle = "The State of the Art in Numerical Analysis",
+ year = "1986",
+ month = "April",
+ editor = "Powell, M.J.D and Iserles, A.",
+ publisher = "Oxford University Press"
+}
\end{chunk}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Petric 71]{Pet71} Petric, S. R. (ed)
Proceedings of the second symposium on Symbolic and
Algebraic Manipulation, March 2325, 1971, Los Angeles, California, ACM Press,
New York, NY 10036, USA, 1971. LCCN QA76.5.S94 1971
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Rich96,
+ author = "Rich, Albert D. and Jeffrey, David J.",
+ title = "Function Evaluation on Branch Cuts",
+ journal = "SIGSAM Bull.",
+ issue_date = "June 1996",
+ volume = "30",
+ number = "2",
+ month = "June",
+ year = "1996",
+ issn = "01635824",
+ pages = "2527",
+ numpages = "3",
+ url = "http://doi.acm.org/10.1145/235699.235704",
+ doi = "10.1145/235699.235704",
+ acmid = "235704",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ abstract = "
+ Once it is decided that a CAS will evaluate multivalued functions on
+ their principal branches, questions arise concerning the branch
+ definitions. The first questions concern the standardization of the
+ positions of the branch cuts. These questions have largely been
+ resolved between the various algebra systems and the numerical
+ libraries, although not completely. In contrast to the computer
+ systems, many mathematical textbooks are much further behind: for
+ example, many popular textbooks still specify that the argument of a
+ complex number lies between 0 and $2\pi$. We do not intend to discuss
+ these first questions here, however. Once the positions of the branch
+ cuts have been fixed, a second set of questions arises concerning the
+ evaluation of functions on their branch cuts."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Pinch 93]{Pin93} Pinch, R.G.E.
``Some Primality Testing Algorithms''
Devlin, Keith (ed.)
Computers and Mathematics November 1993, Vol 40, Number 9 pp12031210
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Patt96,
+ author = "Patton, Charles M.",
+ title = "A Representation of Branchcut Information",
+ journal = "SIGSAM Bull.",
+ issue_date = "June 1996",
+ volume = "30",
+ number = "2",
+ month = "June",
+ year = "1996",
+ issn = "01635824",
+ pages = "2124",
+ numpages = "4",
+ url = "http://doi.acm.org/10.1145/235699.235703",
+ doi = "10.1145/235699.235703",
+ acmid = "235703",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ paper = "Patt96.pdf",
+ abstract = "
+ Handling (possibly) multivalued functions is a problem in all current
+ computer algebra systems. The problem is not an issue of technology.
+ Its solution, however, is tied to a uniform handling of the issues by
+ the mathematics community."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Poll (b)]{Polxx} Poll, Erik
``The type system of Axiom''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Polxx.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Squi91,
+ author = "Squire, Jon S.",
+ title = "Rationale for the Proposed Standard for a Generic Package of
+ Complex Elementary Functions",
+ journal = "Ada Lett.",
+ issue_date = "Fall 1991",
+ volume = "XI",
+ number = "7",
+ month = "September",
+ year = "1991",
+ issn = "10943641",
+ pages = "166179",
+ numpages = "14",
+ url = "http://doi.acm.org/10.1145/123533.123545",
+ doi = "10.1145/123533.123545",
+ acmid = "123545",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ paper = "Squi91.pdf",
+ abstract = "
+ This document provides the background on decisions that were made
+ during the development of the specification for Generic Complex
+ Elementary fuctions. It also rovides some information that was used to
+ develop error bounds, range, domain and definitions of complex
+ elementary functions."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Purtilo 86]{Pur86} Purtilo, J.
``Applications of a software interconnection system in mathematical
problem solving environments'' In Bruce W. Char, editor. Proceedings of the
1986 Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 2123,
ACM Press, New York, NY 10036, USA, 1986. ISBN 0897911997 LCCN QA155.7.E4
A281 1986 ACM order number 505860
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Squi91a,
+ editor = "Squire, Jon S.",
+ title = "Proposed Standard for a Generic Package of Complex
+ Elementary Functions",
+ journal = "Ada Lett.",
+ issue_date = "Fall 1991",
+ volume = "XI",
+ number = "7",
+ month = "September",
+ year = "1991",
+ issn = "10943641",
+ pages = "140165",
+ numpages = "26",
+ url = "http://doi.acm.org/10.1145/123533.123544",
+ doi = "10.1145/123533.123544",
+ acmid = "123544",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ abstract = "
+ This document defines the specification of a generic package of
+ complex elementary functions called Generic Complex Elementary
+ Functions. It does not provide the body of the package."
+}
\end{chunk}
\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Squarefree Decomposition } %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Rainer 14]{Rain14} Joswig, Rainer
``2014: 30+ Years Common Lisp the Language''
\verblispm.de/30ycltl
+\begin{chunk}{axiom.bib}
+@article{Bern97,
+ author = "Bernardin, Laurent",
+ title = "On squarefree factorization of multivariate polynomials over a
+ finite field",
+ journal = "Theoretical Computer Science",
+ volume = "187",
+ number = "12",
+ year = "1997",
+ month = "November",
+ pages = "105116",
keywords = "axiomref",
+ paper = "Bern97.pdf",
+ abstract = "
+ In this paper we present a new deterministic algorithm for computing
+ the squarefree decomposition of multivariate polynomials with
+ coefficients from a finite field.
\end{chunk}
+ Our algorithm is based on Yun's squarefree factorization algorithm
+ for characteristic 0. The new algorithm is more efficient than
+ existing, deterministic algorithms based on Musser's squarefree
+ algorithm
\begin{chunk}{ignore}
\bibitem[Rioboo 03a]{Riob03a} Rioboo, Renaud
``Quelques aspects du calcul exact avec des nombres r\'eels''
Ph.D. Thesis, Laboratoire d'Informatique Th\'eorique et Programmationg
%\verbaxiomdeveloper.org/axiomwebsite/papers/Riob03a.ps
 keywords = "axiomref",
+ We will show that the modular approach presented by Yun has no
+ significant performance advantage over our algorithm. The new
+ algorithm is also simpler to implement and it can rely on any existing
+ GCD algorithm without having to worry about choosing ``good'' evaluation
+ points.
+
+ To demonstrate this, we present some timings using implementations in
+ Maple (Char et al. 1991), where the new algorithm is used for Release
+ 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system
+ known to the author to use and implementation of Yun's modular
+ algorithm mentioned above."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rioboo 03]{Riob03} Rioboo, Renaud
``Towards Faster Real Algebraic Numbers''
J. of Symbolic Computation 36 pp 513533 (2003)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Riob03.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Chez07,
+ author = "Ch\'eze, Guillaume and Lecerf, Gr\'egoire",
+ title = "Lifting and recombination techniques for absolute factorization",
+ journal = "Journal of Complexity",
+ volume = "23",
+ number = "3",
+ year = "2007",
+ month = "June",
+ pages = "380420",
+ paper = "Chez07.pdf",
+ abstract = "
+ In the vein of recent algorithmic advances in polynomial factorization
+ based on lifting and recombination techniques, we present new faster
+ algorithms for computing the absolute factorization of a bivariate
+ polynomial. The running time of our probabilistic algorithm is less
+ than quadratic in the dense size of the polynomial to be factored."
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper presents a new encoding scheme for real algebraic number
manipulations which enhances current Axiom's real closure. Algebraic
manipulations are performed using different instantiations of
subresultantlike algorithms instead of Euclideanlike algorithms.
We use these algorithms to compute polynomial gcds and Bezout
relations, to compute the roots and the signs of algebraic
numbers. This allows us to work in the ring of real algebraic integers
instead of the field of read algebraic numbers avoiding many denominators.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Robidoux 93]{Rob93} Robidoux, Nicolas
``Does Axiom Solve Systems of O.D.E's Like Mathematica?''
July 1993
%\verbaxiomdeveloper.org/axiomwebsite/papers/Rob93.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Lece07,
+ author = "Lecerf, Gr\'egoire",
+ title = "Improved dense multivariate polynomial factorization algorithms",
+ journal = "Journal of Symbolic Computation",
+ volume = "42",
+ number = "4",
+ year = "2007",
+ month = "April",
+ pages = "477494",
+ paper = "Lece07.pdf",
+ abstract = "
+ We present new deterministic and probabilistic algorithms that reduce
+ the factorization of dense polynomials from several variables to one
+ variable. The deterministic algorithm runs in subquadratic time in
+ the dense size of the input polynomial, and the probabilistic
+ algorithm is softly optimal when the number of variables is at least
+ three. We also investigate the reduction from several to two variables
+ and improve the quantitative versions of Bertini's irreducibility theorem."
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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.
+\begin{chunk}{axiom.bib}
+@article{Wang77,
+ author = "Wang, Paul S.",
+ title = "An efficient squarefree decomposition algorithm",
+ journal = "ACM SIGSAM Bulletin",
+ volume = "11",
+ number = "2",
+ year = "1977",
+ month = "May",
+ pages = "46",
+ paper = "Wang77.pdf",
+ abstract = "
+ The concept of polynomial squarefree decomposition is an important one
+ in algebraic computation. The squarefree decomposition process has
+ many uses in computer symbolic computation. A recent survey by D. Yun
+ [3] describes many useful algorithms for this purpose. All of these
+ methods depend on computing the greated common divisor (gcd) of the
+ polynomial to be decomposed and its first derivative (with repect to
+ some variable). In the multivariate case, this gcd computation is
+ nontrivial and dominates the cost for the squarefree decompostion."
+}
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{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rioboo 92]{Rio92} Rioboo, R.
``Real algebraic closure of an ordered field, implementation in Axiom''
In Wang [Wan92], pp206215 ISBN 0897914899 (soft cover)
0897914902 (hard cover) LCCN QA76.95.I59 1992
%\verbaxiomdeveloper.org/axiomwebsite/papers/Rio92.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Wang79,
+ author = "Wang, Paul S. and Trager, Barry M.",
+ title = "New Algorithms for Polynomial SquareFree Decomposition
+ over the Integers",
+ journal = "SIAM Journal on Computing",
+ volume = "8",
+ number = "3",
+ year = "1979",
+ publisher = "Society for Industrial and Applied Mathematics",
+ issn = "00975397",
+ paper = "Wang79.pdf",
+ abstract = "
+ Previously known algorithms for polynomial squarefree decomposition
+ rely on greatest common divisor (gcd) computations over the same
+ coefficient domain where the decomposition is to be performed. In
+ particular, gcd of the given polynomial and its first derivative (with
+ respect to some variable) is obtained to begin with. Application of
+ modular homomorphism and $p$adic construction (multivariate case) or
+ the Chinese remainder algorithm (univariate case) results in new
+ squarefree decomposition algorithms which, generally speaking, take
+ less time than a single gcd between the given polynomial and its first
+ derivative. The key idea is to obtain one or several ``correct''
+ homomorphic images of the desired squarefree decomposition
+ first. This provides information as to how many different squarefree
+ factors there are, their multiplicities and their homomorphic
+ images. Since the multiplicities are known, only the squarefree
+ factors need to be constructed. Thus, these new algorithms are
+ relatively insensitive to the multiplicities of the squarefree factors."
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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
roots of a polynomial. This process is designed to be recursively
used, so the resulting domain of computation is the set of all real
algebraic numbers. An implementation for the real algebraic closure
has been done in Axiom (previously called Scratchpad).
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Roesner 95]{Roe95} Roesner, K. G.
``Verified solutions for parameters of an exact solution for
nonNewtonian liquids using computer algebra'' Zeitschrift fur Angewandte
Mathematik und Physik, 75 (suppl. 2):S435S438, 1995 ISSN 00442267
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@inproceedings{Yun76,
+ author = "Yun, D.Y.Y",
+ title = "On squarefree decomposition algorithms",
+ booktitle = "Proceedings of SYMSAC'76",
+ year = "1976",
+ keywords = "survey",
+ pages = "2635"
+}
\end{chunk}
\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Sage 14]{Sage14} Stein, William
``Sage''
\verbwww.sagemath.org/doc/reference/interfaces/sage/interfaces/axiom.html
 keywords = "axiomref",
+\section{Axiom Citations in the Literature}
\end{chunk}
+\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Salvy 89]{Sal89} Salvy, B.
``Examples of automatic asymptotic expansions''
Technical Report 114,
Inst. Nat. Recherche Inf. Autom., Le Chesnay, France, Dec. 1989 18pp
+\bibitem[ACM 89]{ACM89} ACM, editor
+Proceedings of the ACMSIGSAM 1989 International
+Symposium on Symbolic and Algebraic Computation, ISSAC '89 ACM Press,
+New York, NY 10036, USA, 1989, , LCCN QA76.95.I59
+ year = "1989",
+ isbn = "0897913256",
keywords = "axiomref",

\end{chunk}
\begin{chunk}{ignore}
\bibitem[Salvy 91]{Sal91} Salvy, B.
``Examples of automatic asymptotic expansions''
SIGSAM Bulletin (ACM Special Interest Group on Symbolic and
Algebraic Manipulation), 25(2) pp417
April 1991 CODEN SIGSBZ ISSN 01635824
+\bibitem[ACM 94]{ACM94} ACM, editor
+ISSAC '94. Proceedings of the International
+Symposium on Symbolic and Algebraic Computation. ACM Press, New York, NY,
+10036, USA, 1994, . LCCN QA76.95.I59
+ year = "1994",
+ isbn = "0897916387",
keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Saun80,
 author = "Saunders, B. David",
 title = "A Survey of Available Systems",
 journal = "SIGSAM Bull.",
 issue_date = "November 1980",
 volume = "14",
 number = "4",
 month = "November",
 year = "1980",
 issn = "01635824",
 pages = "1228",
 numpages = "17",
 url = "http://doi.acm.org/10.1145/1089235.1089237",
 doi = "10.1145/1089235.1089237",
 acmid = "1089237",
 publisher = "ACM",
 address = "New York, NY, USA",
 keywords = "axiomref,survey",
 paper = "Saun80.pdf"
}

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Schu 92]{Sch92} Sch\"u, J.
``Implementing des CartanKuranishiTheorems in AXIOM''
Master's diploma thesis (in german), Institut f\"ur Algorithmen und
Kognitive Systeme, Universit\"t Karlsruhe 1992
+@article{Augo91,
+ author = "Augot, D. and Charpin, P. and Sendrier, N.",
+ title = "The miniumum distance of some binary codes via the
+ Newton's identities",
+ journal = "Cohen and Charping [CC91]",
+ year = "1991",
+ pages = "6573",
+ isbn = "0387543031",
+ misc = "3540543031 (Berlin). LCCN QA268.E95 1990",
keywords = "axiomref",
+ paper = "Augo91.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Schwarz 88]{Sch88} Schwarz, F.
``Programming with abstract data types: the symmetry package SPDE
in Scratchpad''
In Jan{\ss}en [Jan88], pp167176, ISBN 3540189289,
0387189289 LCCN QA155.7.E4T74 1988
+\bibitem[Adams 94]{AL94}
+ author = "Adams, William W. and Loustaunau, Philippe",
+ title = "An Introduction to Gr\"obner Bases",
+ year = "1994",
+American Mathematical Society (1994)
+ isbn = "0821838040",
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Schwarz 89]{Sch89} Schwarz, F.
``A factorization algorithm for linear ordinary differential equations''
In ACM [ACM89], pp1725 ISBN 0897913256 LCCN QA76.95.I59 1989
+\bibitem[Andrews 84]{And84}
+ author = "Andrews, George E.",
+ title = "Ramanujan and SCRATCHPAD",
+ year = "1984",
+ pages = "383??",
keywords = "axiomref",
+In Golden and Hussain [GH84]
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Schwarz 91]{Sch91} Schwarz, F.
``Monomial orderings and Gr{\"o}bner bases''
SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic
Manipulation) 2591) pp1023 Jan. 1991 CODEN SIGSBZ ISSN 01635824
+\bibitem[Andrews 88]{And88}
+ author = "Andrews, G. E.",
+ title = "Application of Scratchpad to problems in special functions and
+ combinatorics",
+ year = "1988"
+ pages = "158??",
+ isbn = "3540189289",
keywords = "axiomref",
+In Janssen [Jan88], pages 158?? ISBN
+0387189289 LCCN QA155.7.E4T74
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Seiler 94]{Sei94} Seiler, Werner Markus
``Analysis and Application of the Formal Theory of Partial Differential
Equations''
PhD thesis, School of Physics and Materials, Lancaster University (1994)
\verbwww.mathematik.unikassel.de/~seiler/Papers/Diss/diss.ps.gz
%\verbaxiomdeveloper.org/axiomwebsite/papers/Sei94.pdf
+\bibitem[Anon 91]{Ano91}
+ author = "Anonymous",
+ year = "1991,
keywords = "axiomref",
+Proceedings 1991 Annual Conference, American Society for
+Engineering Education. Challenges of a Changing World. ASEE, Washington, DC
+ 2 vol.
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
An introduction to the formal theory of partial differential equations
is given emphasizing the properties of involutive symbols and
equations. An algorithm to complete any differential equation to an
involutive one is presented. For an involutive equation possible
values for the number of arbitrary functions in its general solution
are determined. The existence and uniqueness of solutions for analytic
equations is proven. Applications of these results include an
analysis of symmetry and reduction methods and a study of gauge
systems. It is show that the Dirac algorithm for systems with
constraints is closely related to the completion of the equation of
motion to an involutive equation. Specific examples treated comprise
the YangMills Equations, Einstein Equations, complete and Jacobian
systems, and some special models in two and three dimensions. To
facilitate the involved tedious computations an environment for
geometric approaches to differential equations has been developed in
the computer algebra system Axiom. The appendices contain among others
brief introductions into CartenK\"ahler Theory and JanetRiquier Theory.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Seiler 94a]{Sei94a} Seiler, W.M.
``Completion to involution in AXIOM''
in Calmet [Cal94] pp103104
+\bibitem[Anon 92]{Ano92}
+ author = "Anonymous",
+ year = "1992",
keywords = "axiomref",
+Programming environments for highlevel scientific problem solving.
+IFIP TC2/WG 2.5 working conference. IFIP Transactions. A Computer Science
+and Technology, A2:??, CODEN ITATEC. ISSN 09265473
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Sieler 94b]{Sei94b} Seiler, W.M.
``Pseudo differential operators and integrable systems in AXIOM''
Computer Physics Communications, 79(2) pp329340 April 1994 CODEN CPHCBZ
ISSN 00104655
%\verbaxiomdeveloper.org/axiomwebsite/papers/Sei94b.pdf
+\bibitem[Anono 95]{Ano95}
+ author =Anonymous
keywords = "axiomref",
+ year = "1995",
+GAMM 94 annual meeting. Zeitschrift fur Angewandte Mathematik und
+Physik, 75 (suppl. 2), CODEN ZAMMAX, ISSN 00442267
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}
+\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Seiler 95]{Sei95} Seiler, W.M.
``Applying AXIOM to partial differential equations''
Internal Report 9517, Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik
1995
%\verbaxiomdeveloper.org/axiomwebsite/papers/Sei95.pdf
+\begin{chunk}{axiom.bib}
+@article{Bacl14,
+ author = "Baclawski, Krystian",
+ title = "SPAD language type checker",
+ journal = "unknown",
+ year = "2014",
+ url = "http://github.com/cahirwpz/phd",
keywords = "axiomref",
+ abstract = "
+ The project aims to deliver a new type checker for SPAD language.
+ Several improvements over current type checker are planned.
+ \begin{itemize}
+ \item introduce better type inference
+ \item introduce modern language constructs
+ \item produce understandable diagnostic messages
+ \item eliminate well known bugs in the type system
+ \item find new type errors
+ \end{itemize}"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Seiler 95b]{SC95} Seiler, W.M.; Calmet, J.
``JET  An Axiom Environment for Geometric Computations with Differential
Equations''
%\verbaxiomdeveloper.org/axiomwebsite/papers/SC95.pdf
+\bibitem[Blair 70]{BGJ70}
+ author = "Blair, Fred W and Griesmer, James H. and Jenks, Richard D.",
+ title = "An interactive facility for symbolic mathematics",
+ year = "1970",
+ pages = "394419",
keywords = "axiomref",
+Proc. International Computing Symposium, Bonn, Germany,
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Seiler 97]{Sei97} Seiler, Werner M.
``Computer Algebra and Differential Equations: An Overview''
\verbwww.mathematik.unikassel.di/~seiler/Papers/Postscript/CADERep.ps.gz
+\bibitem[Blair 70a]{BJ70}
+ author = "Blair, Fred W. and Jenks, Richard D.",
+ title = "LPL: LISP programming language",
+ year = "1970",
keywords = "axiomref",
+IBM Research Report, RC3062 Sept
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present an informal overview of a number of approaches to
differential equations which are popular in computer algebra. This
includes symmetry and completion theory, local analysis, differential
ideal and Galois theory, dynamical systems and numerical analysis. A
large bibliography is provided.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Seiler (a)]{Seixx} Seiler, W.M.
``DETools: A Library for Differential Equations''
\verbiakswww.ira.uka.de/iakscalmet/werner/werner.html
+\begin{chunk}{axiom.bib}
+\bibitem[Broadbery 95]{BGDW95}
+ author = "Broadbery, P. A. and G{\'o}mezD{\'\i}az, T. and Watt, S. M.",
+ title = "On the Implementation of Dynamic Evaluation",
+ year = "1995",
+ pages = "7784",
keywords = "axiomref",
+ isbn = "0897916999",
+ url = "http://pdf.aminer.org/000/449/014/on_the_implementation_of_dynamic_evaluation.pdf",
+ paper = "BGDW95.pdf",
+ abstract = "
+ Dynamic evaluation is a technique for producing multiple results
+ according to a decision tree which evolves with program execution.
+ Sometimes it is desired to produce results for all possible branches
+ in the decision tree, while on other occasions, it may be sufficient
+ to compute a single result which satisfies certain properties. This
+ techinique finds use in computer algebra where computing the correct
+ result depends on recognizing and properly handling special cases of
+ parameters. In previous work, programs using dynamic evaluation have
+ explored all branches of decision trees by repeating the computations
+ prior to decision points.
+
+ This paper presents two new implementations of dynamic evaluation
+ which avoid recomputing intermediate results. The first approach uses
+ Scheme ``continuations'' to record state for resuming program
+ execution. The second implementation uses the Unix ``fork'' operation
+ to form new processes to explore alternative branches in parallel."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Shannon 88]{SS88} Shannon, D.; Sweedler, M.
``Using Gr{\"o}bner bases to determine algebra
membership, split surjective algebra homomorphisms determine birational
equivalence''
Journal of Symbolic Computation 6(23) pp267273
Oct.Dec. 1988 CODEN JSYCEH ISSN 07477171
+\begin{chunk}{axiom.bib}
+\bibitem[Boehm 89]{Boe89}
+@inproceedings{Boe89,
+ author = "Boehm, HansJ.",
+ title = "Type Inference in the Presence of Type Abstraction",
+ year = "1989",
+ pages = "192206",
keywords = "axiomref",
+ url = "http://www.acm.org/pubs/citations/proceedings/pldi/73141/p192boehm",
+ paper = "Boe89.pdf",
+ booktitle = "ACM SIGPLAN Notices",
+ volume = "24",
+ number = "7",
+ month = "July",
+ abstract = "
+ A number of recent programming language designs incorporate a type
+ checking system based on the GirardReynolds polymorphic
+ $\lambda$calculus. This allows the construction of general purpose,
+ reusable software without sacrificing compiletime type checking. A
+ major factor constraining the implementation of these languages is the
+ difficulty of automatically inferring the lengthy type information
+ that is otherwise required if full use is made of these
+ languages. There is no known algorithm to solve any natural and fully
+ general formulation of the ``type inference'' problem. One very
+ reasonable formulation of the problem is known to be undecidable.
+
+ Here we define a restricted version of the type inference problem and
+ present an efficient algorithm for its solution. We argue that the
+ restriction is sufficiently weak to be unobtrusive in practice."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Sit 89]{Sit89} Sit, W.Y.
``On Goldman's algorithm for solving firstorder multinomial
autonomous systems'' In Mora [Mor89], pp386395 ISBN 3540510834
LCCN QA268.A35 1998 Conference held jointly with ISSAC '88
+\begin{chunk}{axiom.bib}
+@inproceedings{BHGM04,
+ author = "Boulton, Richard and Hardy, Ruth and Gottliebsen, Hanne
+ and Martin, Ursula",
+ title = "Design verification for control engineering",
+ year = "2004",
+ month = "April",
+ booktitle = "Proc 4th Int. Conf. on Integrated Formal Methods",
keywords = "axiomref",
+ abstract = "
+ We introduce control engineering as a new domain of application for
+ formal methods. We discuss design verification, drawing attention to
+ the role played by diagrammatic evaluation criteria involving numeric
+ plots of a design, such as Nichols and Bode plots. We show that
+ symbolic computation and computational logic can be used to discharge
+ these criteria and provide symbolic, automated, and very general
+ alternatives to these standard numeric tests. We illustrate our work
+ with reference to a standard reference model drawn from military
+ avionics."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Sit 92]{Sit92} Sit, W.Y.
``An algorithm for solving parametric linear systems''
Journal of Symbolic Computations, 13(4) pp353394, April 1992 CODEN JSYCEH
ISSN 07477171
\verbwww.sciencedirect.com/science/article/pii/S0747717108801046/pdf
\verb?md5=00aa65e18e6ea5c4a008c8dfdfcd4b83&
\verbpid=1s2.0S0747717108801046main.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Sit92.pdf
+\bibitem[Boulanger 91]{Bou91}
+ author = "Boulanger, JeanLouis",
+ title = "Etude de la compilation de scratchpad 2",
+ year = "1991",
+ month = "September",
+Rapport de DEA Universite dl lille 1
keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present a theoretical foundation for studying parametric systesm of
linear equations and prove an efficient algorithm for identifying all
parametric values (including degnerate cases) for which the system is
consistent. The algorithm gives a small set of regimes where for each
regime, the solutions of the specialized systems may be given
uniformly. For homogeneous linear systems, or for systems were the
right hand side is arbitrary, this small set is irredunant. We discuss
in detail practical issues concerning implementations, with particular
emphasis on simplification of results. Examples are given based on a
close implementation of the algorithm in SCRATCHPAD II. We also give a
complexity analysis of the Gaussian elimination method and compare
that with our algorithm.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Sit 06]{Sit06} Sit, Emil
``Tools for Repeatable Research''
\verbwww.emilsit.net/blog/archives/toolsforrepeatableresearch
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Bou93a,
+ author = "Boulanger, JeanLouis",
+ title = "Axiom, language fonctionnel \`a d\'evelopement objet",
+ year = "1993",
+ month = "October",
+ paper = "Bou93a.pdf",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Smedley 92]{Sme92} Smedley, Trevor J.
``Using pictorial and object oriented programming for computer algebra''
In Hal Berghel et al., editors. Applied computing 
technologicial challenges of the 199s: proceedings of the 1992 ACM/SIGAPP
Symposium on Applied Computing, Kansas City Convention Center, March 13, 1992
pp12431247. ACM Press, New York, NY 10036, USA, 1992. ISBN 089791502X
LCCN QA76.76.A65 S95 1992
+\begin{chunk}{axiom.bib}
+@misc{Bou93b,
+ author = "Boulanger, JeanLouis",
+ title = "AXIOM, A Functional Language with Object Oriented Development",
+ year = "1993",
+ paper = "Bou93b.pdf",
keywords = "axiomref",
+ abstract = "
+ We present in this paper, a study about the computer algebra system
+ Axiom, which gives us many very interesting Software engineering
+ concepts. This language is a functional language with an Object
+ Oriented Development. This feature is very important for modeling the
+ mathematical world (Hierarchy) and provides a running with
+ mathematical sense. (All objects are functions). We present many
+ problems of running and development in Axiom. We can note that Aiom is
+ the only system of this category."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Smith 07]{SDJ07} Smith, Jacob; Dos Reis, Gabriel; Jarvi, Jaakko
``Algorithmic differentiation in Axiom''
ACM SIGSAM ISSAC Proceedings 2007 Waterloo, Canada 2007 pp347354
ISBN 9781595937438
%\verbaxiomdeveloper.org/axiomwebsite/papers/SDJ07.pdf
+\bibitem[Boulanger 94]{Bou94}
+ author = "Boulanger, J.L.",
+ title = "Object Oriented Method for Axiom",
+ year = "1995",
+ month = "February",
+ pages = "3341",
+ paper = "Bou94.pdf",
+ACM SIGPLAN Notices, 30(2) CODEN SINODQ ISSN 03621340
keywords = "axiomref",
+ abstract = "
+ Axiom is a very powerful computer algebra system which combines two
+ language paradigms (functional and OOP). Mathematical world is complex
+ and mathematicians use abstraction to design it. This paper presents
+ some aspects of the object oriented development in Axiom. The Axiom
+ programming is based on several new tools for object oriented
+ development, it uses two levels of class and some operations such that
+ {\sl coerce}, {\sl retract}, or {\sl convert} which permit the type
+ evolution. These notions introduce the concept of multiview."
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper describes the design and implementation of an algorithmic
differentiation framework in the Axiom computer algebra system. Our
implementation works by transformations on Spad programs at the level
of the typed abstract syntax tree.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[SSC92]{SSC92}.
``Algorithmic Methods For Lie Pseudogroups''
In N. Ibragimov, M. Torrisi and A. Valenti, editors, Proc. Modern Group
Analysis: Advanced Analytical and Computational Methods in Mathematical
Physics, pp337344, Acireale (Italy), 1992 Kluwer, Dordrecht 1993
\verbiakswww.ira.uka.de/iakscalmet/werner/Papers/Acireale92.ps.gz
+\bibitem[Bronstein 87]{Bro87}
+ author = "Bronstein, Manuel",
+ title = "Integration of Algebraic and Mixed Functions",
+ year = "1987",
+in [Wit87], p18
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[SSV87]{SSV87} Senechaud, P.; Siebert, F.; Villard G.
``Scratchpad II: Pr{\'e}sentation d'un nouveau langage de calcul formel''
Technical Report 640M, TIM 3 (IMAG), Grenoble, France, Feb 1987
+\bibitem[Bronstein 89]{Bro89}
+ author= "Bronstein, M.",
+ title = "Simplification of real elementary functions",
+ year = "1989",
+ pages = "207211",
+ isbn = "0897913256",
+ACM [ACM89] pages LCCN QA76.95.I59 1989
keywords = "axiomref",
+ abstract = "
+ We describe an algorithm, based on Risch's real structure theorem, that
+ determines explicitly all the algebraic relations among a given set of
+ real elementary functions. We also provide examples from its
+ implementation that illustrate the advantages over the use of complex
+ logarithms and exponentials."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Steele]{Steele} Steele, Guy L.; Gabriel, Richard P.
``The Evolution of Lisp''
\verbwww.dreamsongs.com/Files/HOPL2Uncut.pdf
+\begin{chunk}{axiom.bib}
+\bibitem[Bronstein 91a]{Bro91a}
+@inproceedings{Bron91a,
+ author = "Bronstein, M.",
+ title = "The Risch Differential Equation on an Algebraic Curve",
+ booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation",
+ series = "ISSAC'91",
+ year = "1991",
+ pages = "241246",
+ isbn = "0897914376",
+ publisher = "ACM, NY",
keywords = "axiomref",
+ paper = "Bro91a.pdf",
+ abstract = "
+ We present a new rational algorithm for solving Risch differential
+ equations over algebraic curves. This algorithm can also be used to
+ solve $n^{th}$order linear ordinary differential equations with
+ coefficients in an algebraic extension of the rational functions. In
+ the general (``mixed function'') case, this algorithm finds the
+ denominator of any solution of the equation."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Sutor 85]{Sut85} Sutor, R.S.
``The Scratchpad II computer algebra language and system''
In Buchberger and Caviness [BC85], pp3233 ISBN 0387159835 (vol. 1),
0387159843 (vol. 2) LCCN QA155.7.E4 E86 1985 Two volumes.
+\bibitem[Bronstein 91c]{Bro91c}
+ author = "Bronstein, Manuel",
+ title = "Computer Algebra and Indefinite Integrals",
+ year = "1991",
+ paper = "Bro91c.pdf",
+in Computer Aided Proofs in Analysis, K.R. Meyers et al. (eds)
+SpringerVerlag, NY (1991)
keywords = "axiomref",

+ abstract = "
+ We give an overview, from an analytical point of view, of decision
+ procedures for determining whether an elementary function has an
+ elementary function has an elementary antiderivative. We give examples
+ of algebraic functions which are integrable and nonintegrable in
+ closed form, and mention the current implementation of various computer
+ algebra systems."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Sutor 87a]{SJ87a} Sutor, R. S.; Jenks, R. D.
``The type inference and coercion facilities in
the Scratchpad II interpreter'' In Wexelblat [Wex87], pp5663
ISBN 0897912357 LCCN QA76.7.S54 v22:7 SIGPLAN Notices, v22 n7 (July 1987)
%\verbaxiomdeveloper.org/axiomwebsite/papers/SJ87a.pdf
+\bibitem[Bronstein 92]{Bro92}
+ author = "Bronstein, M.",
+ title = "Linear Ordinary Differential Equations: Breaking Through the
+ Order 2 Barrier",
+ year = "1992",
+ url =
+ "http://wwwsop.inria.fr/cafe/Manuel.Bronstein/publications/issac92.ps.gz",
+ paper = "Bro92.pdf",
keywords = "axiomref",

+ abstract = "
+ A major subproblem for algorithms that either factor ordinary linear
+ differential equations or compute their closed form solutions is to
+ find their solutions $y$ which satisfy $y^{'}/y \in \overline{K}(x)$
+ where $K$ is the constant field for the coefficients of the equation.
+ While a decision procedure for this subproblem was known in the
+ $19^{th}$ century, it requires factoring polynomials over
+ $\overline{K}$ and has not been implemented in full generality. We
+ present here an efficient algorithm for this subproblem, which has
+ been implemented in the AXIOM computer algebra system for equations of
+ arbitrary order over arbitrary fields of characteristic 0. This
+ algorithm never needs to compute with the individual complex
+ singularities of the equation, and algebraic numbers are added only
+ when they appear in the potential solutions. Implementation of the
+ complete Singer algorithm for $n=2,3$ based on this building block is
+ in progress."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Sutor 87b]{Su87} Sutor, Robert S.
``The Scratchpad II Computer Algebra System. Using and
Programming the Interpreter''
IBM Course presentation slide deck Spring 1987
+\bibitem[Bronstein 93]{Bro93}
+ author = "Bronstein, Manuel (ed)",
+ year = "1993",
+ month = "July"
+ isbn = "0897916042",
+ISSAC'93: proceedings of the 1993 International Symposium on Symbolic
+and Algebraic Computation, Kiev, Ukraine,
+ACM Press New York, NY 10036, USA, ISBN
+LCCN QA76.95 I59 1993 ACM order number 505930
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Sutor 87c]{SJ87c} Sutor, Robert S.; Jenks, Richard
``The type inference and coercion facilities
in the Scratchpad II interpreter''
Research report RC 12595 (\#56575),
IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1987, 11pp
%\verbaxiomdeveloper.org/axiomwebsite/papers/SJ87c.pdf
+\bibitem[Brunelli 08]{Brun08}
+ author = "Brunelli, J.C.",
+ title = "Streams and Lazy Evaluation Applied to Integrable Models",
+ year = "2008",
+ url = "http://arxiv.org/PS_cache/nlin/pdf/0408/0408058v1.pdf",
+ paper = "Brun08.pdf",
keywords = "axiomref",
+ abstract = "
+ Computer algebra procedures to manipulate pseudodifferential
+ operators are implemented to perform calculations with integrable
+ models. We use lazy evaluation and streams to represent and operate
+ with pseudodifferential operators. No order of truncation is needed
+ since terms are produced on demand. We give a series of concrete
+ examples using the computer algebra language MAPLE."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The Scratchpad II system is an abstract datatype programming language,
a compiler for the language, a library of packages of polymorphic
functions and parameterized abstract datatypes, and an interpreter
that provides sophisticated type inference and coercion facilities.
Although originally designed for the implementation of symbolic
mathematical algorithms, Scratchpad II is a general purpose
programming language. This paper discusses aspects of the
implementation of the intepreter and how it attempts to provide a user
friendly and relatively weakly typed front end for the strongly typed
programming language.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Sutor 88]{Su88} Sutor, Robert S.
``A guide to programming in the scratchpad 2 interpreter''
IBM Manual, March 1988
+\bibitem[Bronstein 93]{BS93}
+ author = "Bronstein, Manuel and Salvy, Bruno",
+ title = "Full Partial Fraction Decomposition of Rational Functions",
+ year = "1993",
+ pages = "157160",
+ isbn = "0897916042",
+In Bronstein [Bro93] LCCN QA76.95 I59 1993
keywords = "axiomref",
\end{chunk}
\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Thompson 00]{Tho00} Thompson, Simon
``Logic and dependent types in the Aldor Computer Algebra System''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Tho00.pdf
+\begin{chunk}{axiom.bib}
+@misc{Bro92a,
+ author = "Bronstein, Manuel",
+ title = "Integration and Differential Equations in Computer Algebra",
+ year = "1992",
+ url = "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.576",
+ paper = "Bro92a.pdf",
keywords = "axiomref",
+ abstract = "
+ We describe in this paper how the problems of computing indefinite
+ integrals and solving linear ordinary differential equations in closed
+ form are now solved by computer algebra systems. After a brief review
+ of the mathematical history of those problems, we outline the two
+ major algorithms for them (respectively the Risch and Singer
+ algorithms) and the recent improvements on those algorithms which has
+ allowed them to be implemented."
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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
type system with {\sl type evaluation} reported elsewhere. The logic
is used to provide an axiomatisation of a number of familiar Aldor
categories as well as a type of vectors.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Thompson (a)]{TTxx} Thompson, Simon; Timochouk, Leonid
``The Aldor\\ language''
%\verbaxiomdeveloper.org/axiomwebsite/papers/TTxx.pdf
+\bibitem[Beneke 94]{BS94}
+ author = "Beneke, T. and Schwippert, W.",
+ title = "Doubletrack into the future: MathCAD will gain new users with
+ Standard and Plus versions",
+ year = "1994",
+ month = "July",
+ pages = "107110",
keywords = "axiomref",
+Elektronik, 43(15) CODEN EKRKAR ISSN 00135658
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper introduces the \verbAldor language, which is a
functional programming language with dependent types and a powerful,
typebased, overloading mechanism. The language is built on a subset
of Aldor, the 'library compiler' language for the Axiom computer
algebra system. \verbAldor is designed with the intention of
incorporating logical reasoning into computer algebra computations.

The paper contains a formal account of the semantics and type system
of \verbAldor; a general discussion of overloading and how the
overloading in \verbAldor fits into the general scheme; examples
of logic within \verbAldor and notes on the implementation of the
system.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Touratier 98]{Tou98} Touratier, Emmanuel
``Etude du typage dans le syst\`eme de calcul scientifique Aldor''
Universit\'e de Limoges 1998
%\verbaxiomdeveloper.org/axiomwebsite/papers/Tou98.pdf
+\bibitem[Bronstein 97a]{Bro97a}
+ author = "Bronstein, Manuel and Weil, JacquesArthur",
+ title = "On Symmetric Powers of Differential Operators",
+ series = "ISSAC'97",
+ year = "1997",
+ pages = "156163",
keywords = "axiomref",
+ url =
+ "http://wwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html"
+ paper = "Bro97a.pdf",
+ publisher = "ACM, NY",
+ abstract = "
+ We present alternative algorithms for computing symmetric powers of
+ linear ordinary differential operators. Our algorithms are applicable
+ to operators with coefficients in arbitrary integral domains and
+ become faster than the traditional methods for symmetric powers of
+ sufficiently large order, or over sufficiently complicated coefficient
+ domains. The basic ideas are also applicable to other computations
+ involving cyclic vector techniques, such as exterior powers of
+ differential or difference operators."
\end{chunk}
\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[van der Hoeven 14]{JvdH14} van der Hoeven, Joris
``Computer algebra systems and TeXmacs''
\verbwww.texmacs.org/tmweb/plugins/cas.en.html
 keywords = "axiomref",
+\bibitem[Borwein 00]{Bor00}
+ author = "Borwein, Jonathan",
+ title = "Multimedia tools for communicating mathematics",
+ year = "2000",
+ pages = "58",
+ isbn = "3540424504",
+ publisher = "SpringerVerlag",
+ keywords = "axiomref"
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Hoei94,
 author = "{van Hoeij}, M.",
 title = "An algorithm for computing an integral basis in an algebraic function field",
+@article{BT94,
+ author = "Brown, R. and Tonks, A.",
+ title = "Calculations with simplicial and cubical groups in AXIOM",
journal = "Journal of Symbolic Computation",
 volume = "18",
 number = "4",
+ volume = "17",
+ number = "2",
+ pages = "159179",
year = "1994",
 pages = "353363",
 issn = "07477171",
 keywords = "axiomref",
 paper = "Hoei94.pdf"
+ month = "February",
+ misc = "CODEN JSYCEH ISSN 07477171",
+ keywords = "axiomref"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Algorithms for computing integral bases of an algebraic function field
are implemented in some computer algebra systems. They are used e.g.
for the integration of algebraic functions. The method used by Maple
5.2 and AXIOM is given by Trager in [Trag84]. He adapted an algorithm
of Ford and Zassenhaus [Ford, 1978], that computes the ring of
integers in an algebraic number field, to the case of a function field.

It turns out that using algebraic geometry one can write a faster
algorithm. The method we will give is based on Puiseux expansions.
One cas see this as a variant on the Coates' algorithm as it is
described in [Davenport, 1981]. Some difficulties in computing with
Puiseux expansions can be avoided using a sharp bound for the number
of terms required which will be given in Section 3. In Section 5 we
derive which denominator is needed in the integral basis. Using this
result 'intermediate expression swell' can be avoided.

The Puiseux expansions generally introduce algebraic extensions. These
extensions will not appear in the resulting integral basis.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@misc{Hoei08,
 author = "{van Hoeij}, Mark and Novocin, Andrew",
 title = "A Reduction Algorithm for Algebraic Function Fields",
 year = "2008",
 month = "April",
 url = "http://andy.novocin.com/pro/algext.pdf",
 paper = "Hoei08.pdf"
}

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
Computer algebra systesm often produce large expressions involving
complicated algebraic numbers. In this paper we study variations of
the {\tt polred} algorithm that can often be used to find better
representations for algebraic numbers. The main new algorithm
presented here is an algorithm that treats the same problem for the
function field case.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Vasconcelos 99]{Vas99} Vasconcelos, Wolmer
``Computational Methods in Commutative Algebra and Algebraic Geometry''
Springer, Algorithms and Computation in Mathematics, Vol 2 1999
ISBN 3540213112
 keywords = "axiomref",

\end{chunk}

\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Wang 89]{Wan89} Wang, D.
``A program for computing the Liapunov functions and Liapunov
constants in Scratchpad II''
SIGSAM Bulletin (ACM Special Interest Group
on Symbolic and Algebraic Manipulation), 23(4) pp2531, Oct. 1989,
CODEN SIGSBZ ISSN 01635824
+@misc{Brow95,
+ author = "Brown, Ronald and Dreckmann, Winfried",
+ title = "Domains of data and domains of terms in AXIOM",
+ year = "1995",
keywords = "axiomref",
+ paper = "DB95.pdf",
+ abstract = "
+ The main new concept we wish to illustrate in this paper is a
+ distinction between ``domains of data'' and ``domains of terms'', and
+ its use in the programming of certain mathematical structures.
+ Although this distinction is implicit in much of the programming work
+ that has gone into the construction of Axiom categories and domains,
+ we believe that a formalisation of this is new, that standards and
+ conventions are necessary and will be useful in various other
+ contexts. We shall show how this concept may be used for the coding of
+ free categories and groupoids on directed graphs."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wang 91]{Wan91} Wang, Dongming
``Mechanical manipulation for a class of differential systems''
Journal of Symbolic Computation, 12(2) pp233254 Aug. 1991
CODEN JSYCEH ISSN 07477171
+\bibitem[Buchberger 85]{BC85} Buchberger, Bruno and Caviness, Bob F. (eds)
+EUROCAL '85: European Conference on Computer Algebra, Linz, Austria,
+LLCN QA155.7.E4 E86
+ isbn = "0387159835, 0387159843",
+ year = "1985",
+ month = "April",
+ publisher = "SpringerVerlag, Berlin, Germany",
keywords = "axiomref",
+ misc = "Lecture Notes in Computer Science, Vol 204",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wang 92]{Wan92} Wang, Paul S. (ed)
International System Symposium on Symbolic and
Algebraic Computation 92 ACM Press, New York, NY 10036, USA, 1992
ISBN 0897914899 (soft cover), 0897914902 (hard cover),
LCCN QA76.95.I59 1992
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Buh05,
+ author = "Buhl, Soren L.",
+ title = "Some Reflections on Integrating a Computer Algebra System in R",
+ year = "2005",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watanabe 90]{WN90} Watanabe, Shunro; Nagata, Morio; (ed)
ISSAC '90 Proceedings of the
International Symposium on Symbolic and Algebraic Computation ACM Press,
New York, NY, 10036, USA. 1990 ISBN 0897914015 LCCN QA76.95.I57 1990
+\bibitem[Burge 91]{Burg91}
+ author = "Burge, W.H.",
+ title = "Scratchpad and the RogersRamanujan identities",
+ year = "1991",
+ pages = "189190",
+ isbn = "0897914376",
keywords = "axiomref",
+ abstract = "
+ This note sketches the part played by Scratchpad in obtaining new
+ proofs of Euler's theorem and the RogersRamanujan Identities."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 85]{Wat85} Watt, Stephen
``Bounded Parallelism in Computer Algebra''
PhD Thesis, University of Waterloo
\verbwww.csd.uwo.ca/~watt/pub/reprints/1985smwphd.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@techreport{BW87,
+ author = "Burge, W. and Watt, S.",
+ title = "Infinite structures in SCRATCHPAD II",
+ year = "1987",
+ institution = "IBM Research",
+ type = "Technical Report",
+ number = "RC 12794",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 86]{Wat86} Watt, S.M.; Della Dora, J.
``Algebra Snapshot: Linear Ordinary Differential Operators''
Scratchpad II Newsletter: Vol 1 Num 2 (Jan 1986)
\verbwww.csd.uwo.ca/~watt/pub/reprints/1986snewslodo.pdf
+\bibitem[Burge 87a]{BWM87}
+ author = "Burge, William H. and Watt, Stephen M. and Morrison, Scott C.",
+ title = "Streams and Power Series",
+ year = "1987",
+ pages = "912",
keywords = "axiomref",
+in [Wit87], pp912
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 87]{Wat87} Watt, Stephen
``Domains and Subdomains in Scratchpad II''
in [Wit87], pp35
+\bibitem[Burge 89]{BW89}
+ author = "Burge, W. H. and Watt, S. M.",
+ title = "Infinite structures in Scratchpad II",
+ year = "1989",
+ pages = "138148",
+ isbn = "3540515178",
keywords = "axiomref",
+in Davenport [Dav89], LCCN QA155.7.E4E86 1987
\end{chunk}
+\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Watt 87a]{WB87} Watt, Stephen M.; Burge, William H.
``Mapping as First Class Objects''
in [Wit87], pp1317
+\bibitem[Calmet 94]{Cal94} Calmet, J. (ed)
+Rhine Workshop on Computer Algebra, Proceedings.
+Universit{\"a}t Karsruhe, Karlsruhe, Germany 1994
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 89]{Wat89} Watt, S. M.
``A fixed point method for power series computation''
In Gianni [Gia89], pp206217 ISBN 3540510842 LCCN QA76.95.I57
1988 Conference held jointly with AAECC6
+\bibitem[Camion 92]{CCM92}
+ author = "Camion, Paul and Courteau, Bernard and Montpetit, Andre",
+ title = "A combinatorial problem in Hamming Graphs and its solution
+ in Scratchpad",
+ year = "1992",
+ month = "January",
keywords = "axiomref",
+Rapports de recherche 1586, Institut National de Recherche en
+Informatique et en Automatique, Le Chesnay, France, 12pp
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 90]{WJST90} Watt, S.M.; Jenks, R.D.; Sutor, R.S.; Trager B.M.
``The Scratchpad II type system: Domains and subdomains''
in A.M. Miola, editor Computing Tools
for Scientific Problem Solving, Academic Press, New York, 1990
+\bibitem[Caprotti 00]{CCR00}
+ author = "Caprotti, Olga and Cohen, Arjeh M. and Riem, Manfred",
+ title = "Java Phrasebooks for Computer Algebra and Automated Deduction",
+ url = "http://www.sigsam.org/bulletin/articles/132/paper8.pdf",
+ paper = "CCR00.pdf",
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 91]{Wat91} Watt, Stephen M. (ed)
Proceedings of the 1991 International Symposium on
Symbolic and Algebraic Computation, ISSAC'91, July 1517, 1991, Bonn, Germany,
ACM Press, New York, NY 10036, USA, 1991 ISBN 0897914376
LCCN QA76.95.I59 1991
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{CC99,
+ author = "Capriotti, O. and Carlisle, D.",
+ title = "OpenMath and MathML: Semantic Mark Up for Mathematics",
+ year = "1999",
+ url = "http://www.acm.org/crossroads/xrds62/openmath.html",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 94a]{Wat94a} Watt, Stephen M.; Dooley, S.S.; Morrison, S.C.;
Steinback, J.M.; Sutor, R.S.
``A\# User's Guide''
Version 1.0.0 O($\epsilon{}^1$) June 8, 1994
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Capr99,
+ author = "Capriotti, Olga and Cohen, Arjeh M. and Cuypers, Hans and
+ Sterk, Hans",
+ title = "OpenMath Technology for Interactive Mathematical Documents",
+ year = "2002",
+ pages = "5166",
+ publisher = "SpringerVerlag, Berlin, Germany",
+ url = "http://www.win.tue.nl/~hansc/lisbon.pdf",
+ paper = "Capr99.pdf",
+ misc = "in Multimedia Tools for Communicating Mathematics",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 94b]{Wat94} Watt, Stephen M.; Broadbery, Peter A.;
Dooley, Samuel S.; Iglio, Pietro
``A First Report on the A\# Compiler (including benchmarks)''
IBM Research Report RC19529 (85075) May 12, 1994
%\verbaxiomdeveloper.org/axiomwebsite/papers/Wat94.pdf
 keywords = "axiomref",
+\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",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 94c]{Wat94c} Watt, Stephen M.
``A\# Language Reference Version 0.35''
IBM Research Division Technical Report RC19530 May 1994
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Chu85,
+ author = "Chudnovsky, D.V and Chudnovsky, G.V.",
+ title = "Elliptic Curve Calculations in Scratchpad II",
+ year = "1985",
+ institution = "Mathematics Dept., IBM Research",
+ type = "Scratchpad II Newsletter 1 (1)",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 95]{Wat95} Watt, S.M.; Broadbery, P.A.; Dooley, S.S.; Iglio, P.
Steinbach, J.M.; Morrison, S.C.; Sutor, R.S.
``AXIOM Library Compiler Users Guide''
The Numerical Algorithms Group (NAG) Ltd, 1994
+\bibitem[Chudnovsky 87]{Chu87}
+ author = "Chudnovsky, D.V and Chudnovsky, G.V.",
+ title = "New Analytic Methods of Polynomial Root Finding",
+ year = "1987",
+ pages = "2",
keywords = "axiomref",
+in [Wit87]
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Watt 01]{Wat01} Watt, Stephen M.; Broadbery, Peter A.; Iglio, Pietro;
Morrison, Scott C.; Steinbach, Jonathan M.
``FOAM: A First Order Abstract Machine Version 0.35''
IBM T. J. Watson Research Center (2001)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Wat01.pdf
+\bibitem[Chudnovsky 89]{Chu89}
+ author = "Chudnovsky, D.V. and Chudnovsky, G.V.",
+ title = "The computation of classical constants",
+ year = "1989",
+ month = "November",
+ pages = "81788182",
keywords = "axiomref",
+Proc. Natl. Acad. Sci. USA Vol 86
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Weber 92]{Webe92} Weber, Andreas
``Type Systems for Computer Algebra''
\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber92a.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe92.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@proceedings{CJ86,
+ editor = "Chudnovsky, David and Jenks, Richard",
+ title = "Computers in Mathematics",
+ year = "1986",
+ month = "July",
+ isbn = "0824783417",
+ note = "International Conference on Computers and Mathematics",
+ publisher = "Marcel Dekker, Inc",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
An important feature of modern computer algebra systems is the support
of a rich type system with the possibility of type inference. Basic
features of such a type system are polymorphism and coercion between
types. Recently the use of ordersorted rewrite systems was proposed
as a general framework. We will give a quite simple example of a
family of types arising in computer algebra whose coercion relations
cannot be captured by a finite set of firstorder rewrite rules.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Weber 92b]{Webe92b} Weber, Andreas
``Structuring the Type System of a Computer Algebra System''
\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber92a.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe92b.pdf
+\begin{chunk}{axiom.bib}
+@misc{Cohe03,
+ author = "Cohen, Arjeh and Cuypers, M. and Barreiro, Hans and
+ Reinaldo, Ernesto and Sterk, Hans",
+ title = "Interactive Mathematical Documents on the Web",
+ year = "2003",
+ pages = "289306",
+ editor = "Joswig, M. and Takayma, N.",
+ publisher = "SpringerVerlag, Berlin, Germany",
keywords = "axiomref",
+ misc = "in Algebra, Geometry and Software Systems"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Most existing computer algebra systems are pure symbol manipulating
systems without language support for the occuring types. This is
mainly due to the fact taht the occurring types are much more
complicated than in traditional programming languages. In the last
decade the study of type systems has become an active area of
research. We will give a proposal for a type system showing that
several problems for a type system of a symbolic computation system
can be solved by using results of this research. We will also provide
a variety of examples which will show some of the problems that remain
and that will require further research.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Weber 93b]{Webe93b} Weber, Andreas
``Type Systems for Computer Algebra''
\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber93b.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe93b.pdf
+\bibitem[Cohen 91]{CC91} Cohen, G.; Charpin, P.; (ed)
+EUROCODE '90 International Symposium on
+Coding Theory and Applications Proceedings. SpringerVerlag, Berlin, Germany
+/ Heidelberg, Germany / London, UK / etc., 1991 ISBN 0387543031
+(New York), 3540543031 (Berlin), LCCN QA268.E95 1990
keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We study type systems for computer algebra systems, which frequently
correspond to the ``pragmatically developed'' typing constructs used
in AXIOM. A central concept is that of {\sl type classes} which
correspond to AXIOM categories. We will show that types can be
syntactically described as terms of a regular ordersorted signature
if no type parameters are allowed. Using results obtained for the
functional programming language Haskell we will show that the problem
of {\sl type inference} is decidable. This result still holds if
higherorder functions are present and {\sl parametric polymorphism}
is used. These additional typing constructs are useful for further
extensions of existing computer algebra systems: These typing concepts
can be used to implement category theoretic constructs and there are
many well known constructive interactions between category theory and
algebra. \end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Weber 94]{Web94} Weber, Andreas
``Algorithms for Type Inference with Coercions''
ISSAC 94 ACM 0897916387/94/0007
%\verbaxiomdeveloper.org/axiomwebsite/papers/Web94.pdf
+\bibitem[Conrad (a)]{CFMPxxa}
+ 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",
+ 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}
\begin{adjustwidth}{2.5em}{0pt}
This paper presents algorithms that perform a type inference for a
type system occurring in the context of computer algebra. The type
system permits various classes of coercions between types and the
algorithms are complete for the precisely defined system, which can be
seen as a formal description of an important subset of the type system
supported by the computer algebra program Axiom.
+\begin{chunk}{axiom.bib}
+@misc{CFMPxxb,
+ author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra",
+ title = "Mathematical Use Cases lead naturally to nonstandard Inheritance
+ Relationships: How to make them accessible in a mainstream language?",
+ paper = "CFMPxxb.pdf",
+ keywords = "axiomref",
+ abstract = "
+ Conceptually there is a strong correspondence between Mathematical
+ Reasoning and ObjectOriented techniques. We investigate how the ideas
+ of Method Renaming, Dynamic Inheritance and Interclassing can be used
+ to strengthen this relationship. A discussion is initiated concerning
+ the feasibility of each of these features."
+}
Previously only algorithms for much more restricted cases of coercions
have been described or the frameworks used have been so general that
the corresponding type inference problems were known to be
undecidable.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Weber 95]{Webe95} Weber, A.
``On coherence in computer algebra''
\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber94e.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe95.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Cuyp10,
+ author = "Cuypers, Hans and Hendriks, Maxim and Knopper, Jan Willem",
+ title = "Interactive Geometry inside MathDox",
+ year = "2010",
+ url = "http://www.win.tue.nl/~hansc/MathDox_and_InterGeo_paper.pdf",
+ paper = "Cuyp10",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Modern computer algebra systems (e.g. AXIOM) support a rich type
system including parameterized data types and the possibility of
implicit coercions between types. In such a type system it will be
frequently the case that there are different ways of building
coercions between types. An important requirement is that all
coercions between two types coincide, a property which is called {\sl
coherence}. We will prove a coherence theorem for a formal type system
having several possibilities of coercions covering many important
examples. Moreover, we will give some informal reasoning why the
formally defined restrictions can be satisfied by an actual system.
\end{adjustwidth}
+\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Weber 96]{Webe96} Weber, Andreas
``Computing Radical Expressions for Roots of Unity''
\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber96a.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe96.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@inproceedings{Dalm97,
+ author = {Dalmas, St\'ephane and Ga\"etano, Marc and Watt, Stephen},
+ title = "An OpenMath 1.0 Implementation",
+ booktitle = "Proc. 1997 Int. Symp. on Symbolic and Algebraic Computation",
+ series = "ISSAC'97",
+ year = "1997",
+ isbn = "0897918754",
+ location = "Kihei, Maui, Hawaii, USA",
+ pages = "241248",
+ numpages = "8",
+ url = "http://doi.acm.org/10.1145/258726.258794",
+ doi = "10.1145/258726.258794",
+ acmid = "258794",
+ publisher = "ACM, New York, NY USA",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present an improvement of an algorithm given by Gauss to compute a
radical expression for a $p$th root of unity. The time complexity of
the algorithm is $O(p^3m^6log p)$, where $m$ is the largest prime
factor of $p1$.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Weber 99]{Webe99} Weber, Andreas
``Solving Cyclotomic Polynomials by Radical Expressions''
\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/
\verbWeberA/WeberKeckeisen99a.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe99.pdf
+\bibitem[Dalmas 92]{Dal92} Dalmas, S.
+``A polymorphic functional language applied to symbolic computation''
+In Wang [Wan92] pp369375 ISBN 0897914899 (soft cover) 0897914902
+(hard cover) LCCN QA76.95.I59 1992
keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe a Maple package that allows the solution of cyclotomic
polynomials by radical expressions. We provide a function that is an
extension of the Maple {\sl solve} command. The major algorithmic
ingredient of the package is an improvement of a method due to Gauss
which gives radical expressions for roots of unity. We will give a
summary for computations up to degree 100, which could be done within
a few hours of cpu time on a standard workstation.
\end{adjustwidth}
+\begin{chunk}{axiom.bib}
+@misc{Daly88,
+ author = "Daly, Timothy",
+ title = "Axiom in an Educational Setting, Axiom course slide deck",
+ year = "1988",
+ month = "January",
+ keywords = "axiomref"
+}
\begin{chunk}{ignore}
\bibitem[WeiJiang 12]{WJ12} WeiJiang
``Top free algebra System''
\verbweijiang.com/it/software/topfreealgebrasystembyemathematicabyemaple
+\end{chunk}
+
+\begin{chunk}{ignore}TPDHERE
+\bibitem[Daly 02]{Dal02} Daly, Timothy
+``Axiom as open source''
+SIGSAM Bulletin (ACM Special Interest Group
+on Symbolic and Algebraic Manipulation) 36(1) pp28?? March 2002
+CODEN SIGSBZ ISSN 01635824
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wester 99]{Wes99} Wester, Michael J.
``Computer Algebra Systems''
John Wiley and Sons 1999 ISBN 0471983535
+\bibitem[Daly 03]{Dal03} Daly, Timothy
+``The Axiom Wiki Website''
+\verbaxiom.axiomdeveloper.org
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wexelblat 87]{Wex87} Wexelblat, Richard L. (ed)
Proceedings of the SIGPLAN '87 Symposium on
Interpreter and Interpretive Techniques, St. Paul, Minnesota, June 2426, 1987
ACM Press, New York, NY 10036, USA, 1987 ISBN 0897912357
LCCN QA76.7.S54 v22:7 SIGPLAN Notices, vol 22, no 7 (July 1987)
+\bibitem[Daly 06]{Dal06} Daly, Timothy
+``Axiom Volume 1: Tutorial''
+Lulu, Inc. 860 Aviation Parkway,
+Suite 300, Morrisville, NC 27560 USA, 2006 ISBN 141166597X 287pp
+\verbwww.lulu.com/content/190827
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wityak 87]{Wit87} Wityak, Sandra
``Scratchpad II Newsletter''
Volume 2, Number 1, Nov 1987
+\bibitem[Daly 09]{Dal09} Daly, Timothy
+``The Axiom Literate Documentation''
+\verbaxiomdeveloper.org/axiomwebsite/documentation.html
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[WWW1]{WWW1}.
Software Preservation Group
\verbwww.softwarepresentation.org/projects/LISP/common_lisp_family
+\bibitem[Daly 13]{Dal13} Daly, Timothy
+``Literate Programming in the Large''
+April 89, 2013 Portland Oregon
+\verbconf.writethedocs.org
+\verbdaly.axiomdeveloper.org
+\verbwww.youtube.com/watch?v=Av0PQDVTP4A
keywords = "axiomref",
\end{chunk}
\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Yap 00]{Yap00} Yap, Chee Keng
``Fundamental Problems of Algorithmic Algebra''
Oxford University Press (2000) ISBN0195125169
+\bibitem[Davenport 79a]{Dav79a} Davenport, J.H.
+``What can SCRATCHPAD/370 do?''
+VM/370 SPAD.SCRIPTS August 24, 1979 SPAD.SCRIPT
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Yapp 07]{Yapp07} Yapp, Clifford; Hebisch, Waldek; Kaminski, Kai
``Literate Programming Tools Implemented in ANSI Common Lisp''
\verbbrlcad.org/~starseeker/clwebv0.8.lisp.pamphlet
+\bibitem[Davenport 80]{Dav80} Davenport, J.H.; Jenks, R.D.
+``MODLISP  an Introduction''
+Proc LISP80, 1980, and IBM RC8357 Oct 1980
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Yun 83]{Yun83} Yun, David Y.Y.
``Computer Algebra and Complex Analysis''
Computational Aspects of Complex Analysis pp379393
D. Reidel Publishing Company H. Werner et. al. (eds.)
+\bibitem[Davenport 84]{DGJ84} Davenport, J.; Gianni, P.; Jenks, R.;
+Miller, V.; Morrison, S.; Rothstein, M.; Sundaresan, C.; Sutor, R.;
+Trager, B.
+``Scratchpad''
+Mathematical Sciences Department, IBM Thomas Watson Research Center 1984
keywords = "axiomref",
\end{chunk}
\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Zen92]{Zen92} Zenger, Ch.
``Gr{\"o}bnerbasen f{\"u}r Differentialformen und ihre
Implementierung in AXIOM''
Diplomarbeit, Universit{\"a}t Karlsruhe,
Karlsruhe, Germany, 1992
+\bibitem[Davenport 84a]{Dav84a} Davenport, James H.
+``A New Algebra System''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav84a.pdf
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Zip92]{Zip92} Zippel, Richard
``Algebraic Computation''
(unpublished) Cornell University Ithaca, NY Sept 1992
+\bibitem[Davenport 85]{Dav85} Davenport, James H.
+``The LISP/VM Foundation of Scratchpad II''
+The Scratchpad II Newsletter, Volume 1, Number 1, September 1, 1985
+IBM Corporation, Yorktown Heights, NY
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Zwi92]{Zwi92} Zwillinger, Daniel
``Handbook of Integration''
Jones and Bartlett, 1992, ISBN 0867202939
+\bibitem[Davenport 88]{DST88} Davenport, J.H.; Siret, Y.; Tournier, E.
+Computer Algebra: Systems and Algorithms for Algebraic Computation.
+Academic Press, New York, NY, USA, 1988, ISBN 0122042329
+\verbstaff.bath.ac.uk/masjhd/masternew.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/DST88.pdf
keywords = "axiomref",
\end{chunk}
\section{Axiom Citations of External Sources}

\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{axiom.bib}
@article{Abla98,
 author = "Ablamowicz, Rafal",
 title = "Spinor Representations of Clifford Algebras: A Symbolic Approach",
 journal = "Computer Physics Communications",
 volume = "115",
 number = "23",
 month = "December",
 year = "1998",
 pages = "510535"
}

\end{chunk}
\begin{chunk}{axiom.bib}
@article{Abra06,
 author = "Abramov, Sergey A.",
 title = "In Memory of Manuel Bronstein",
 journal = "Programming and Computer Software",
 volume = "32",
 number = "1",
 pages = "5658",
 publisher = "Pleiades Publishing Inc",
 year = "2006",
 paper = "Abra06.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Davenport 14]{Dav14} Davenport, James H.
+``Computer Algebra textbook''
+\verbstaff.bath.ac.uk/masjhd/JHDCA.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav14.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Abramowitz 64]{AS64} Abramowitz, Milton; Stegun, Irene A.
``Handbook of Mathematical Functions''
(1964) Dover Publications, NY ISBN 0486612724
+\bibitem[Davenport 89]{Dav89} Davenport, J.H. (ed)
+EUROCAL '87 European Conference on Computer Algebra Proceedings
+SpringerVerlag, Berlin, Germany / Heidelberg, Germany / London,
+UK / etc., 1989 ISBN 3540515178 LCCN QA155.7.E4E86 1987
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Abramowitz 68]{AS68} Abramowitz M; Stegun I A
``Handbook of Mathematical Functions''
Dover Publications. (1968)
+\bibitem[Davenport 90]{DT90} Davenport, J. H.; Trager, B. M.
+``Scratchpad's view of algebra I: Basic commutative algebra''
+In Miola [Mio90], pp4054. ISBN 0387525319 (New York),
+3540525319 (Berlin). LCCN QA76.9.S88I576 1990 also in AXIOM Technical
+Report, ATR/1, NAG Ltd., Oxford, 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Altm05,
 author = "Altmann, Simon L.",
 title = "Rotations, Quaternions, and Double Groups",
 publisher = "Dover Publications, Inc.",
 year = "2005",
 isbn = "0486445186"
+@inproceedings{Dave91,
+ author = "Davenport, J. H. and Gianni, P. and Trager, B. M.",
+ title = "Scratchpad's View of Algebra II:
+ A Categorical View of Factorization",
+ booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation",
+ series = "ISSAC '91",
+ year = "1991",
+ isbn = "0897914376",
+ location = "Bonn, West Germany",
+ pages = "3238",
+ numpages = "7",
+ url = "http://doi.acm.org/10.1145/120694.120699",
+ doi = "10.1145/120694.120699",
+ acmid = "120699",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ keywords = "axiomref",
+ paper = "Dave91.pdf",
+ abstract = "
+ This paper explains how Scratchpad solves the problem of presenting a
+ categorical view of factorization in unique factorization domains,
+ i.e. a view which can be propagated by functors such as
+ SparseUnivariatePolynomial or Fraction. This is not easy, as the
+ constructive version of the classical concept of
+ UniqueFactorizationDomain cannot be so propagated. The solution
+ adopted is based largely on Seidenberg's conditions (F) and (P), but
+ there are several additional points that have to be borne in mind to
+ produce reasonably efficient algorithms in the required generality.
+
+ The consequence of the algorithms and interfaces presented in this
+ paper is that Scratchpad can factorize in any extension of the
+ integers or finite fields by any combination of polynomial, fraction
+ and algebraic extensions: a capability far more general than any other
+ computer algebra system possesses. The solution is not perfect: for
+ example we cannot use these general constructions to factorize
+ polyinmoals in $\overline{Z[\sqrt{5}]}[x]$ since the domain
+ $Z[\sqrt{5}]$ is not a unique factorization domain, even though
+ $\overline{Z[\sqrt{5}]}$ is, since it is a field. Of course, we can
+ factor polynomials in $\overline{Z}[\sqrt{5}][x]$"
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ames 77]{Ames77} Ames W F
``Nonlinear Partial Differential Equations in Engineering''
Academic Press (2nd Edition). (1977)
+\bibitem[Davenport 92]{DGT92} Davenport, J. H.;, Gianni, P.; Trager, B. M.
+``Scratchpad's view of algebra II: A categorical view of factorization''
+Technical Report TR4/92 (ATR/2) (NP2491), Numerical Algorithms Group, Inc.,
+Downer's Grove, IL, USA and Oxford, UK, December 1992
+\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Amos 86]{Amos86} Amos D E
``Algorithm 644: A Portable Package for Bessel Functions of a Complex
Argument and Nonnegative Order''
ACM Trans. Math. Softw. 12 265273. (1986)
+\bibitem[Davenport 92a]{Dav92a} Davenport, J. H.
+``The AXIOM system''
+AXIOM Technical Report TR5/92 (ATR/3)
+(NP2492) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and
+Oxford, UK, December 1992
+\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Anderson 00]{And00} Anderson, Edward
``Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem''
LAPACK Working Note 150, University of Tennessee, UTCS00454,
December 4, 2000.
+\bibitem[Davenport 92b]{Dav92b} Davenport, J. H.
+``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
+\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav92b.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}
\begin{chunk}{ignore}
\bibitem[Anthony 82]{ACH82} Anthony G T; Cox M G; Hayes J G
``DASL  Data Approximation Subroutine Library''
National Physical Laboratory. (1982)
+\bibitem[Davenport 92c]{DT92} Davenport, J. H.; Trager, B. M.
+``Scratchpad's view of algebra I: Basic commutative algebra''
+DISCO 90 Capri, Italy April 1990 ISBN 0387525319 pp4054
+Technical Report TR3/92 (ATR/1)(NP2490), Numerical
+Algorithms Group, Inc., Downer's Grove, IL, USA and Oxford, UK,
+December 1992.
+\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Arnon 88]{Arno88} Arno, D.S.; MIgnotte, M.
``On Mechanical Quantifier Elimination for Elementary Algebra and Geometry''
J. Symbolic Computation 5, 237259 (1988)
\verbhttp://www.sciencedirect.com/science/article/pii/S0747717188800142/
\verbpdf?md5=62052077d84e6078cc024bc8e29c23c1&
\verbpid=1s2.0S0747717188800142main.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Arno88.pdf
+\bibitem[Davenport 93]{Dav93} Davenport, J. H.
+``Primality testing revisited''
+Technical Report TR2/93 (ATR/6)(NP2556) Numerical Algorithms Group, Inc.,
+Downer's Grove, IL, USA and Oxford, UK, August 1993
+\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We give solutions to two problems of elementary algebra and geometry:
(1) find conditions on real numbers $p$, $q$, and $r$ so that the
polynomial function $f(x)=x^4+px^2+qx+r$ is nonnegative for all real
$x$ and (2) find conditions on real numbers $a$, $b$, and $c$ so that
the ellipse $\frac{(xe)^2}{q^2}+\frac{y^2}{b^2}1=0$ lies inside the
unit circle $y^2+x^21=0$. Our solutions are obtained by following the
basic outline of the method of quantifier elimination by cylindrical
algebraic decomposition (Collins, 1975), but we have developed, and
have been considerably aided by, modified versions of certain of its
steps. We have found three equally simple but not obviously equivalent
solutions for the first problem, illustrating the difficulty of
obtaining unique ``simplest'' solutions to quantifier elimination
problems of elementary algebra and geometry.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{Aubr99,
 author = "Aubry, Phillippe and Lazard, Daniel and {Moreno Maza}, Marc",
 title = "On the Theories of Triangular Sets",
 year = "1999",
 pages = "105124",
 journal = "Journal of Symbolic Computation",
 volume = "28",
 url = "http://www.csd.uwo.ca/~moreno/Publications/AubryLazardMorenoMaza1999JSC.pdf",
 papers = "Aubr99.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Davenport (a)]{DFxx} Davenport, James; Faure, Christ\'ele
+``The Unknown in Computer Algebra''
+\verbaxiomwiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/DFxx.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}
\begin{adjustwidth}{2.5em}{0pt}
Different notions of triangular sets are presented. The relationship
between these notions are studied. The main result is that four
different existing notions of {\sl good} triangular sets are
equivalent.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Aubry 96]{Aub96} Aubry, Philippe; Maza, Marc Moreno
``Triangular Sets for Solving Polynomial Systems: a Comparison of Four Methods''
\verbwww.lip6.fr/lip6/reports/1997/lip6.1997.009.ps.gz
%\verbaxiomdeveloper.org/axiomwebsite/papers/Aub96.ps
+\bibitem[Davenport 00]{Dav00} Davenport, James
+``13th OpenMath Meeting''
+James H. Davenport
+``A New Algebra System''
+May 1984
+\verbxml.coverpages.org/openmath13.html
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav00.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Four methods for solving polynomial systems by means of triangular
sets are presented and implemented in a unified way. These methods are
those of Wu, Lazard, Kalkbrener, and Wang. They are compared on
various examples with emphasis on efficiency, conciseness and
legibility of the outputs.
\end{adjustwidth}

\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Bailey 66]{Bai66} Bailey P B
``SturmLiouville Eigenvalues via a Phase Function''
SIAM J. Appl. Math . 14 242249. (1966)
+\bibitem[Davenport 12]{Dav12} Davenport, J.H.
+``Computer Algebra''
+\verbstaff.bath.ac.uk/masjhd/JHDCA.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Baker 96]{BGM96} Baker, George A.; GravesMorris, Peter
``Pade Approximants''
Cambridge University Press, March 1996 ISBN 9870521450072
+\bibitem[Davenport (b)]{DSTxx} Davenport, J. H.; Siret; Tournier
+``Computer Algebra'' \hfill
+\verbstaff.bath.ac.uk/masjhd/masternew.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Baker 10]{Ba10} Baker, Martin
``3D World Simulation''
\verbwww.euclideanspace.com
+\bibitem[Dewar 94]{Dew94} Dewar, M. C.
+``Manipulating Fortran Code in AXIOM and the AXIOMNAG Link''
+Proceedings of the Workshop on Symbolic and Numeric Computing, ed by Apiola, H.
+and Laine, M. and Valkeila, E. pp112 University of Helsinki, Finland (1994)
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Bake14,
 author = "Baker, Martin",
 title = "Axiom Architecture",
 year = "2014",
 url = "http://www.euclideanspace.com/prog/scratchpad/internals/ccode"
+@misc{Dewa,
+ author = "Dewar, Mike",
+ title = "OpenMath: An Overview",
+ url = "http://www.sigsam.org/bulletin/articles/132/paper1.pdf",
+ paper = "Dewa.pdf",
+ keywords = "axiomref"
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Banks 68]{BK68} Banks D O; Kurowski I
``Computation of Eigenvalues of Singular SturmLiouville Systems''
Math. Computing. 22 304310. (1968)
+\bibitem[Dicrescenzo 89]{DD89} Dicrescenzo, C.; Duval, D.
+``Algebraic extensions and algebraic closure in Scratchpad II''
+In Gianni [Gia89], pp440446 ISBN 3540510842
+LCCN QA76.95.I57 1998 Conference held jointly with AAECC6
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bard 74]{Bard74} Bard Y
``Nonlinear Parameter Estimation''
Academic Press. 1974
+\bibitem[Dingle 94]{Din94} Dingle, Adam; Fateman, Richard
+``Branch Cuts in Computer Algebra''
+1994 ISSAC, Oxford (UK), July 1994
+\verbwww.cs.berkeley.edu/~fateman/papers/ding.ps
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Din94.pdf
+ keywords = "axiomref",
+ abstract = "
+ Many standard functions, such as the logarithms and square root
+ functions, cannot be defined continuously on the complex
+ plane. Mistaken assumptions about the properties of these functions
+ lead computer algebra systems into various conundrums. We discuss how
+ they can manipulate such functions in a useful fashion."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Barrodale 73]{BR73} Barrodale I; Roberts F D K
``An Improved Algorithm for Discrete $ll_1$ Linear Approximation''
SIAM J. Numer. Anal. 10 839848. (1973)
+\bibitem[DLMF]{DLMF}.
+``Digital Library of Mathematical Functions''
+\verbdlmf.nist.gov/software/#T1
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Barrodale 74]{BR74} Barrodale I; Roberts F D K
``Solution of an Overdetermined System of Equations in the $ll_1norm$.''
Comm. ACM. 17, 6 319320. (1974)
+\bibitem[Dooley 99]{Doo99} Dooley, Sam editor.
+ISSAC 99: July 2931, 1999, Simon Fraser University,
+Vancouver, BC, Canada: proceedings of the 1999 International Symposium on
+Symbolic and Algebraic Computation. ACM Press, New York, NY 10036, USA, 1999.
+ISBN 1581130732 LCCN QA76.95.I57 1999
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Beauzamy 92]{Bea92} Beauzamy, Bernard
``Products of polynomials and a priori estimates for
coefficients in polynomial decompositions: a sharp result''
J. Symbolic Computation (1992) 13, 463472
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bea92.pdf

+\bibitem[Dos Reis 12]{DR12} Dos Reis, Gabriel
+``A System for Axiomatic Programming''
+Proc. Conf. on Intelligent Computer Mathematics, Springer (2012)
+\verbwww.axiomatics.org/~gdr/liz/cicm2012.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/DR12.pdf
+ keywords = "axiomref",
+ abstract = "
+ We present the design and implementation of a system for axiomatic
+ programming, and its application to mathematical software
+ construction. Key novelties include a direct support for userdefined
+ axioms establishing local equality between types, and overload
+ resolution based on equational theories and userdefined local
+ axioms. We illustrate uses of axioms, and their organization into
+ concepts, in structured generic programming as practiced in
+ computational mathematical systems."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Beauzamy 93]{Bea93} Beauzamy, Bernard; Trevisan, Vilmar;
Wang, Paul S.
``Polynomial Factorization: Sharp Bounds, Efficient Algorithms''
J. Symbolic Computation (1993) 15, 393413
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bea93.pdf
+\bibitem[Doye 97]{Doy97} Doye, Nicolas James
+``Order Sorted Computer Algebra and Coercions''
+Ph.D. Thesis University of Bath 1997
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Doy97.pdf
+ keywords = "axiomref",
+ abstract = "
+ Computer algebra systems are large collections of routines for solving
+ mathematical problems algorithmically, efficiently and above all,
+ symbolically. The more advanced and rigorous computer algebra systems
+ (for example, Axiom) use the concept of strong types based on
+ ordersorted algebra and category theory to ensure that operations are
+ only applied to expressions when they ``make sense''.
\end{chunk}
+ In cases where Axiom uses notions which are not covered by current
+ mathematics we shall present new mathematics which will allow us to
+ prove that all such cases are reducible to cases covered by the
+ current theory. On the other hand, we shall also point out all the
+ cases where Axiom deviates undesirably from the mathematical ideal.
+ Furthermore we shall propose solutions to these deviations.
\begin{chunk}{axiom.bib}
@article{Bert95,
 author = "Bertrand, Laurent",
 title = "Computing a hyperelliptic integral using arithmetic in the jacobian of the curve",
 journal = "Applicable Algebra in Engineering, Communication and Computing",
 volume = "6",
 pages = "275298",
 year = "1995"
}
+ Strongly typed systems (especially of mathematics) become unusable
+ unless the system can change the type in a way a user expects. We wish
+ any change expected by a user to be automated, ``natural'', and
+ unique. ``Coercions'' are normally viewed as ``natural type changing
+ maps''. This thesis shall rigorously define the word ``coercion'' in
+ the context of computer algebra systems.
\end{chunk}
+ We shall list some assumptions so that we may prove new results so
+ that all coercions are unique. This concept is called ``coherence''.
\begin{adjustwidth}{2.5em}{0pt}
In this paper, we describe an efficient algorithm for computing an
elementary antiderivative of an algebraic function defined on a
hyperelliptic curve. Our algorithm combines B.M. Trager's integration
algorithm and a technique for computing in the Jacobian of a
hyperelliptic curve introduced by D.G. Cantor. Our method has been
implemented and successfully compared to Trager's general algorithm.
\end{adjustwidth}
+ We shall give an algorithm for automatically creating all coercions in
+ type system which adheres to a set of assumptions. We shall prove that
+ this is an algorithm and that it always returns a coercion when one
+ exists. Finally, we present a demonstration implementation of this
+ automated coerion algorithm in Axiom."
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Berzins 87]{BBG87} Berzins M; Brankin R W; Gladwell I.
``Design of the Stiff Integrators in the NAG Library''
Technical Report. TR14/87 NAG. (1987)
+\bibitem[Doye 99]{Doy99} Doye, Nicolas J.
+``Automated coercion for Axiom''
+In Dooley [Doo99], pp229235
+ISBN 1581130732 LCCN QA76.95.I57 1999 ACM Press
+\verbwww.acm.org/citation.cfm?id=309944
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Berzins 90]{Ber90} Berzins M
``Developments in the NAG Library Software for Parabolic Equations''
Scientific Software Systems. (ed J C Mason and M G Cox)
Chapman and Hall. 5972. (1990)
+\bibitem[Dominguez 01]{DR01} Dom\'inguez, C\'esar; Rubio, Julio
+``Modeling Inheritance as Coercion in a Symbolic Computation System''
+ISSAC 2001 ACM 1581134177/01/0007
+%\verbaxiomdeveloper.org/axiomwebsite/papers/DR01.pdf
+ keywords = "axiomref",
+ abstract = "
+ In this paper the analysis of the data structures used in a symbolic
+ computation system, called Kenzo, is undertaken. We deal with the
+ specification of the inheritance relationship since Kenzo is an
+ objectoriented system, written in CLOS, the Common Lisp Object
+ System. We focus on a particular case, namely the relationship between
+ simplicial sets and chain complexes, showing how the ordersorted
+ algebraic specifications formalisms can be adapted, through the
+ ``inheritance as coercion'' metaphor, in order to model this Kenzo
+ fragment."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Birkhoff 62]{BR62} Birkhoff, G; Rota, G C
``Ordinary Differential Equations''
Ginn \& Co., Boston and New York. (1962)
+\bibitem[Dunstan 97]{Dun97} Dunstan, Martin and Ursula, Martin and
+ Linton, Steve
+``Embedded Verification Techniques for Computer Algebra Systems''
+Grant citation GR/L48256 Nov 1, 1997Feb 28, 2001
+\verbwww.cs.standrews.ac.uk/research/output/detail?output=ML97.php
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Boyd9 3a]{Boyd93a} Boyd, David W.
``Bounds for the Height of a Factor of a Polynomial in
Terms of Bombieri's Norms: I. The Largest Factor''
J. Symbolic Computation (1993) 16, 115130
%\verbaxiomdeveloper.org/axiomwebsite/Boyd93a.pdf
+\bibitem[Adams 01]{DGKM01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne;
+Kelsey, Tom; Martin, Ursula; Owre, Sam
+``Computer Algebra meets Automated Theorem Proving: Integrating Maple and PVS''
+TPHOLS 2001, Edinburgh
+\verbwww.csl.sri.com/~owre/papers/tphols01/tphols01.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/DGKM01.pdf
+ keywords = "axiomref",
+ abstract = "
+ We describe an interface between version 6 of the Maple computer
+ algebra system with the PVS automated theorem prover. The interface is
+ designed to allow Maple users access to the robust and checkable proof
+ environment of PVS. We also extend this environment by the provision
+ of a library of proof strategies for use in real analysis. We
+ demonstrate examples using the interface and the real analysis
+ library. These examples provide proofs which are both illustrative and
+ applicable to genuine symbolic computation problems."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Boyd 93b]{Boyd93b} Boyd, David W.
``Bounds for the Height of a Factor of a Polynomial in
Terms of Bombieri's Norms: II. The Smallest Factor''
J. Symbolic Computation (1993) 16, 131145
%\verbaxiomdeveloper.org/axiomwebsite/Boyd93b.pdf
+\bibitem[Duval 92]{DJ92} Duval D.; Jung, F.
+``Examples of problem solving using computer algebra''
+IFIP Transactions. A. Computer Science and Technology, A2 pp133141, 143 1992
+CODEN ITATEC. ISSN 09265473
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Braman 02a]{BBM02a} Braman, K.; Byers, R.; Mathias, R.
``The MultiShift QR Algorithm Part I: Maintaining Well Focused Shifts,
and Level 3 Performance''
SIAM Journal of Matrix Analysis, volume 23, pages 929947, 2002.
+\bibitem[Duval 94]{Duv94} Duval, Dominique
+``Symbolic or algebraic computation?''
+Madrid Spain, NAG conference (private copy of paper)
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Braman 02b]{BBM02b} Braman, K.; Byers, R.; Mathias, R.
``The MultiShift QR Algorithm Part II: Aggressive Early Deflation''
SIAM Journal of Matrix Analysis, volume 23, pages 948973, 2002.
+\begin{chunk}{axiom.bib}
+@article{Duva95,
+ author = "Duval, D.",
+ title = "Evaluation dynamique et cl\^oture alg\'ebrique en Axiom",
+ journal = "Journal of Pure and Applied Algebra",
+ volume = "99",
+ year = "1995",
+ pages = "267295.",
+ keywords = "axiomref"
+}
\end{chunk}
+\subsection{E} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Brent 75]{Bre75} Brent, R. P.
``MultiplePrecision ZeroFinding Methods and the Complexity
of Elementary Function Evaluation, Analytic Computational Complexity''
J. F. Traub, Ed., Academic Press, New York 1975, 151176
+\bibitem[Erocal 10]{ES10} Er\"ocal, Burcin; Stein, William
+``The Sage Project''
+\verbwstein.org/papers/icms/icms_2010.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/ES10.pdf
+ keywords = "axiomref",
+ abstract = "
+ Sage is a free, open source, selfcontained distribution of
+ mathematical software, including a large library that provides a
+ unified interface to the components of this distribution. This library
+ also builds on the components of Sage to implement novel algorithms
+ covering a broad range of mathematical functionality from algebraic
+ combinatorics to number theory and arithmetic geometry."
\end{chunk}
+\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Brent 78]{BK78} Brent, R. P.; Kung, H. T.
``Fast Algorithms for Manipulating Formal Power Series''
Journal of the Association for Computing Machinery,
Vol. 25, No. 4, October 1978, 581595
+\bibitem[Fateman 90]{Fat90} Fateman, R. J.
+``Advances and trends in the design and construction of algebraic
+manipulation systems''
+In Watanabe and Nagata [WN90], pp6067 ISBN 0897914015 LCCN QA76.95.I57 1990
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Brigham 73]{Bri73} Brigham E O
``The Fast Fourier Transform''
PrenticeHall. (1973)
+\bibitem[Fateman 05]{Fat05} Fateman, R. J.
+``An incremental approach to building a mathematical expert out of software''
+4/19/2005\hfill
+\verbwww.cs.berkeley.edu/~fateman/papers/axiom.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat05.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Brillhart 69]{Bri69} Brillhart, John
``On the Euler and Bernoulli polynomials''
J. Reine Angew. Math., v. 234, (1969), pp. 4564
+\bibitem[Fateman 06]{Fat06} Fateman, R. J.
+``Building Algebra Systems by Overloading Lisp''
+\verbwww.cs.berkeley.edu/~fateman/generic/overloadsmall.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat06.pdf
+ keywords = "axiomref",
+ abstract = "
+ Some of the earliest computer algebra systems (CAS) looked like
+ overloaded languages of the same era. FORMAC, PL/I FORMAC, Formula
+ Algol, and others each took advantage of a preexisting language base
+ and expanded the notion of a numeric value to include mathematical
+ expressions. Much more recently, perhaps encouraged by the growth in
+ popularity of C++, we have seen a renewal of the use of overloading to
+ implement a CAS.
+
+ This paper makes three points. 1. It is easy to do overloading in
+ Common Lisp, and show how to do it in detail. 2. Overloading per se
+ provides an easy solution to some simple programming problems. We show
+ how it can be used for a ``demonstration'' CAS. Other simple and
+ plausible overloadings interact nicely with this basic system. 3. Not
+ all goes so smoothly: we can view overloading as a case study and
+ perhaps an object lesson since it fails to solve a number of
+ fairlywell articulated and difficult design issues in CAS for which
+ other approaches are preferable."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Brillhart 90]{Bri90} Brillhart, John
``Note on Irreducibility Testing''
Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 13791381
+\bibitem[Faure 00a]{FDN00a} Faure, Christ\'ele; Davenport, James
+``Parameters in Computer Algebra''
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 98a]{Bro98a} Bronstein, M.; Grabmeier, J.; Weispfenning, V. (eds)
``Symbolic Rewriting Techniques''
Progress in Computer Science and Applied Logic 15, BirkhauserVerlag, Basel
ISBN 3764359013 (1998)
+\bibitem[Faure 00b]{FDN00b} Faure, Christ\'ele; Davenport, James;
+Naciri, Hanane
+``Multivalues Computer Algebra''
+ISSN 02496399 Institut National De Recherche en Informatique et en
+Automatique Sept. 2000 No. 4001
+\verbhal.inria.fr/inria00072643/PDF/RR4401.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/FDN00b.pdf
+ keywords = "axiomref",
+ abstract = "
+ One of the main strengths of computer algebra is being able to solve a
+ family of problems with one computation. In order to express not only
+ one problem but a family of problems, one introduces some symbols
+ which are in fact the parameters common to all the problems of the
+ family.
+
+ The user must be able to understand in which way these parameters
+ affect the result when he looks at the answer. Otherwise it may lead
+ to completely wrong calculations, which when used for numerical
+ applications bring nonsensical answers. This is the case in most
+ current Computer Algebra Systems we know because the form of the
+ answer is never explicitly conditioned by the values of the
+ parameters. The user is not even informed that the given answer may be
+ wrong in some cases then computer algebra systems can not be entirely
+ trustworthy. We have introduced multivalued expressions called {\sl
+ conditional} expressions, in which each potential value is associated
+ with a condition on some parameters. This is used, in particular, to
+ capture the situation in integration, where the form of the answer can
+ depend on whether certain quantities are positive, negative or
+ zero. We show that it is also necessary when solving modular linear
+ equations or deducing congruence conditions from complex expressions."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 88]{Bro88} Bronstein, Manual
``The Transcendental Risch Differential Equation''
J. Symbolic Computation (1990) 9, pp4960 Feb 1988
IBM Research Report RC13460 IBM Corp. Yorktown Heights, NY
\verbwww.sciencedirect.com/science/article/pii/S0747717108800065
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro88.pdf
+\bibitem[Fitch 84]{Fit84} Fitch, J. P. (ed)
+EUROSAM '84: International Symposium on Symbolic and
+Algebraic Computation, Cambridge, England, July 911, 1984, volume 174 of
+Lecture Notes in Computer Science. SpringerVerlag, Berlin, Germany /
+Heildelberg, Germany / London, UK / etc., 1984 ISBN 038713350X
+LCCN QA155.7.E4 I57 1984
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present a new rational algorithm for solving Risch differential
equations in towers of transcendental elementary extensions. In
contrast to a recent algorithm by Davenport we do not require a
progressive reduction of the denominators involved, but use weak
normality to obtain a formula for the denominator of a possible
solution. Implementation timings show this approach to be faster than
a Hermitelike reduction.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@techreport{Bron98,
 author = "Bronstein, Manuel",
 title = "The lazy hermite reduction",
 type = "Rapport de Recherche",
 number = "RR3562",
 year = "1998",
 institution = "French Institute for Research in Computer Science",
 paper = "Bron98.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Fitch 93]{Fit93} Fitch, J. (ed)
+Design and Implementation of Symbolic Computation Systems
+International Symposium DISCO '92 Proceedings. SpringerVerlag, Berlin,
+Germany / Heildelberg, Germany / London, UK / etc., 1993. ISBN 0387572724
+(New York), 3540572724 (Berlin). LCCN QA76.9.S88I576 1992
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The Hermite reduction is a symbolic integration technique that reduces
algebraic functions to integrands having only simple affine
poles. While it is very effective in the case of simple radical
extensions, its use in more general algebraic extensions requires the
precomputation of an integral basis, which makes the reduction
impractical for either multiple algebraic extensions or complicated
ground fields. In this paper, we show that the Hermite reduction can
be performed without {\sl a priori} computation of either a primitive
element or integral basis, computing the smallest order necessary for
a particular integrand along the way.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Fogus 11]{Fog11} Fogus, Michael
+``UnConj''
+\verbclojure.com/blog/2011/11/22/unconj.html
+ keywords = "axiomref",
+
+\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Bro98b,
 author = "Bronstein, Manuel",
 title = "Symbolic Integration Tutorial",
 series = "ISSAC'98",
 year = "1998",
 address = "INRIA Sophia Antipolis",
 url = "http://wwwsop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf",
 paper = "Bro98b.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Fortenbacher 90]{For90} Fortenbacher, A.
+``Efficient type inference and coercion in computer algebra''
+In Miola [Mio90], pp5660. ISBN 0387525319 (New York), 3540525319
+(Berlin). LCCN QA76.9.S88I576 1990
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Brown 99]{Brow99} Brown, Christopher W.
``Solution Formula Construction for Truth Invariant CADs''
Ph.D Thesis, Univ. Delaware (1999)
\verbwww.usna.edu/Users/cs/wcbrown/research/thesis.ps.gz
%\verbaxiomdeveloper.org/axiomwebsite/papers/Brow99.pdf
+\bibitem[Fouche 90]{Fou90} Fouche, Francois
+``Une implantation de l'algorithme de Kovacic en Scratchpad''
+Technical report, Institut de Recherche Math{\'{e}}matique Avanc{\'{e}}e''
+Strasbourg, France, 1990 31pp
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The CADbased quantifier elimination algorithm takes a formula from
the elementary theory of real closed fields as input, and constructs a
CAD of the space of the formula's unquantified variables. This
decomposition is truth invariant with respect to the input formula,
meaning that the formula is either identically true or identically
false in each cell of the decomposition. The method determines the
truth of the input formula for each cell of the CAD, and then uses the
CAD to construct a solution formula  a quantifier free formula that
is equivalent to the input formula. This final phase of the algorithm,
the solution formula construction phase, is the focus of this thesis.

An optimal solution formula construction algorithm would be {\sl
complete}  i.e. applicable to any truthinvariant CAD, would be {\sl
efficient}, and would produce {\sl simple} solution formulas. Prior to
this thesis, no method was available with even two of these three
properties. Several algorithms are presented, all addressing problems
related to solution formula construction. In combination, these
provide an efficient and complete method for constructing solution
formulas that are simple in a variety of ways.
+\begin{chunk}{ignore}
+\bibitem[FSF 14]{FSF14} FSF
+``Free Software Directory''
+\verbdirectory.fsf.org/wiki/Axiom
+ keywords = "axiomref",
Algorithms presented in this thesis have been implemented using the
SACLIB library, and integrated into QEPCAD, a SACLIBbased
implementation of quantifier elimination by CAD. Example computations
based on these implementations are discussed.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Brown 02]{Brow02} Brown, Christopher W.
``QEPCAD B  A program for computing with semialgebraic sets using CADs''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Brow02.pdf
+\bibitem[Frisco ]{Fris} Frisco
+``Objectives and Results''
+\verbwww.nag.co.uk/projects/frisco/frisco/node3.htm
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This report introduces QEPCAD B, a program for computing with real
algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD
B both extends and improves upon the QEPCAD system for quantifier
elimination by partial cylindrical algebraic decomposition written by
Hoon Hong in the early 1990s. This paper briefly discusses some of the
improvements in the implementation of CAD and quantifier elimination
vis CAD, and provides somewhat more detail on extensions to the system
that go beyond quantifier elimination. The author is responsible for
most of the extended features of QEPCAD B, but improvements to the
basic CAD implementation and to the SACLIB library on which QEPCAD is
based are the results of many people's work.
\end{adjustwidth}
+\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@article{Burg74,
 author = "William H. Burge",
 title = "Stream Processing Functions",
 year = "1974",
 month = "January",
 journal = "IBM Journal of Research and Development",
 volume = "19",
 issue = "1",
 pages = "1225",
 papers = "Burg74.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Gebauer 86]{GM86} Gebauer, R{\"u}diger; M{\"o}ller, H. Michael
+``Buchberger's algorithm and staggered linear bases''
+In Bruce W. Char, editor. Proceedings of the 1986
+Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 2123, 1986
+Waterloo, Ontario, pp218221 ACM Press, New York, NY 10036, USA, 1986.
+ISBN 0897911997 LCCN QA155.7.E4 A281 1986 ACM order number 505860
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
One principle of structured programming is that a program should be
separated into meaningful independent subprograms, which are then
combined so that the relation of the parts to the whole can be clearly
established. This paper describes several alternative ways to compose
programs. The main method used is to permit the programmer to denote
by an expression the sequence of values taken on by a variable. The
sequence is represented by a function called a stream, which is a
functional analog of a coroutine. The conventional while and for loops
of structured programming may be composed by a technique of stream
processing (analogous to list processing), which results in more
structured programs than the orignals. This technique makes it
possible to structure a program in a natural way into its logically
separate parts, which can then be considered independently.
\end{adjustwidth}

\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Carlson 65]{Car65} Carlson B C
``On Computing Elliptic Integrals and Functions''
J Math Phys. 44 3651. (1965)
+\bibitem[Gebauer 88]{GM88} Gebauer, R.; M{\"o}ller, H. M.
+``On an installation of Buchberger's algorithm''
+Journal of Symbolic Computation, 6(23) pp275286 1988
+CODEN JSYCEH ISSN 07477171
+\verbwww.sciencedirect.com/science/article/pii/S0747717188800488/pdf
+\verb?md5=f6ccf63002ef3bc58aaa92e12ef18980&
+\verbpid=1s2.0S0747717188800488main.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/GM88.pdf
+ keywords = "axiomref",
+ abstract = "
+ Buchberger's algorithm calculates Groebner bases of polynomial
+ ideals. Its efficiency depends strongly on practical criteria for
+ detecting superfluous reductions. Buchberger recommends two
+ criteria. The more important one is interpreted in this paper as a
+ criterion for detecting redundant elements in a basis of a module of
+ syzygies. We present a method for obtaining a reduced, nearly minimal
+ basis of that module. The simple procedure for detecting (redundant
+ syzygies and )superfluous reductions is incorporated now in our
+ installation of Buchberger's algorithm in SCRATCHPAD II and REDUCE
+ 3.3. The paper concludes with statistics stressing the good
+ computational properties of these installations."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Carlson 77a]{Car77a} Carlson B C
``Elliptic Integrals of the First Kind''
SIAM J Math Anal. 8 231242. (1977)
+\begin{chunk}{axiom.bib}
+@book{Gedd92,
+ author = "Geddes, Keith and Czapor, O. and Stephen R. and Labahn, George",
+ title = "Algorithms For Computer Algebra",
+ publisher = "Kluwer Academic Publishers",
+ isbn = "0792392590",
+ month = "September",
+ year = "1992",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Carlson 77b]{Car77b} Carlson B C
``Special Functions of Applied Mathematics''
Academic Press. (1977)
+\bibitem[Gianni 87]{Gia87} Gianni, Patrizia
+``Primary Decomposition of Ideals''
+in [Wit87], pp1213
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Carlson 78]{Car78} Carlson B C,
``Computing Elliptic Integrals by Duplication''
(Preprint) Department of Physics, Iowa State University. (1978)
+\bibitem[Gianni 88]{Gia88} Gianni, Patrizia.; Trager, Barry.;
+Zacharias, Gail.
+``Gr\"obner Bases and Primary Decomposition of Polynomial Ideals''
+J. Symbolic Computation 6, 149167 (1988)
+\verbwww.sciencedirect.com/science/article/pii/S0747717188800403/pdf
+\verb?md5=40c29b67947035884904fd4597ddf710&
+\verbpid=1s2.0S0747717188800403main.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Gia88.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Carlson 88]{Car88} Carlson B C,
``A Table of Elliptic Integrals of the Third Kind''
Math. Comput. 51 267280. (1988)
+\bibitem[Gianni 89a]{Gia89} Gianni, P. (Patrizia) (ed)
+Symbolic and Algebraic Computation.
+International Symposium ISSAC '88, Rome, Italy, July 48, 1988. Proceedings,
+volume 358 of Lecture Notes in Computer Science. SpringerVerlag, Berlin,
+Germany / Heildelberg, Germany / London, UK / etc., 1989. ISBN 3540510842
+LCCN QA76.95.I57 1988 Conference held jointly with AAECC6
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cauchy 1829]{Cau1829} AugustinLux Cauchy
``Exercices de Math\'ematiques Quatri\`eme Ann\'ee. De Bure Fr\`eres''
Paris 1829 (reprinted Oeuvres, II S\'erie, Tome IX,
GauthierVillars, Paris, 1891).
+\bibitem[Gianni 89b]{GM89} Gianni, P.; Mora, T.
+``Algebraic solution of systems of polynomial equations using
+Gr{\"o}bner bases.''
+In Huguet and Poli [HP89], pp247257 ISBN 3540510826 LCCN QA268.A35 1987
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ch\`eze 07]{Chez07} Ch\'eze, Guillaume; Lecerf, Gr\'egoire
``Lifting and recombination techniques for absolute factorization''
Journal of Complexity, VOl 23 Issue 3 June 2007 pp 380420
\verbwww.sciencedirect.com/science/article/pii/S0885064X07000465
%\verbaxiomdeveloper.org/axiomwebsite/papers/Chez07.pdf
+\bibitem[Gil 92]{Gil92} Gil, I.
+``Computation of the Jordan canonical form of a square matrix (using
+the Axiom programming language)''
+In Wang [Wan92], pp138145.
+ISBN 0897914899 (soft cover), 0897914902 (hard cover)
+LCCN QA76.95.I59 1992
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In the vein of recent algorithmic advances in polynomial factorization
based on lifting and recombination techniques, we present new faster
algorithms for computing the absolute factorization of a bivariate
polynomial. The running time of our probabilistic algorithm is less
than quadratic in the dense size of the polynomial to be factored.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Childs 79]{CSDDN79} Childs B; Scott M; Daniel J W; Denman E;
Nelson P (eds)
``Codes for Boundaryvalue Problems in Ordinary Differential Equations''
Lecture Notes in Computer Science. 76 (1979) SpringerVerlag
+\bibitem[GomezDiaz 92]{Gom92} G\'omezD'iaz, Teresa
+``Quelques applications de l`\'evaluation dynamique''
+Ph.D. Thesis L'Universite De Limoges March 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Clausen 89]{Cla89} Clausen, M.; Fortenbacher, A.
``Efficient Solution of Linear Diophantine Equations''
JSC (1989) 8, 201216
+\bibitem[GomezDiaz 93]{Gom93} G\'omezD\'iaz, Teresa
+``Examples of using Dynamic Constructible Closure''
+IMACS Symposium SC1993
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Gom93.pdf
+ keywords = "axiomref",
+ abstract = "
+ We present here some examples of using the ``Dynamic Constructible
+ Closure'' program, which performs automatic case distinction in
+ computations involving parameters over a base field $K$. This program
+ is an application of the ``Dynamic Evaluation'' principle, which
+ generalizes traditional evaluation and was first used to deal with
+ algebraic numbers."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Clenshaw 55]{Cle55} Clenshaw C W,
``A Note on the Summation of Chebyshev Series''
Math. Tables Aids Comput. 9 118120. (1955)
+\bibitem[Goodwin 91]{GBL91} Goodwin, B. M.; Buonopane, R. A.; Lee, A.
+``Using MathCAD in teaching material and energy balance concepts''
+In Anonymous [Ano91], pp345349 (vol. 1) 2 vols.
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Clenshaw 60]{Cle60} Clenshaw C W
``Curve Fitting with a Digital Computer''
Comput. J. 2 170173. (1960)
+\bibitem[Golden 4]{GH84} Golden, V. Ellen; Hussain, M. A. (eds)
+Proceedings of the 1984 MACSYMA Users' Conference:
+Schenectady, New York, July 2325, 1984, General Electric,
+Schenectady, NY, USA, 1984
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Clenshaw 62]{Cle62} Clenshaw C W
``Mathematical Tables. Chebyshev Series for Mathematical Functions''
HMSO. (1962)
+\bibitem[Gonnet 96]{Gon96} Gonnet, Gaston H.
+``Official verion 1.0 of the Meta Content Dictionary''
+\verbwww.inf.ethz.ch/personal/gonnet/ContDict/Meta
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cline 84]{CR84} Cline A K; Renka R L,
``A Storageefficient Method for Construction of a Thiessen Triangulation''
Rocky Mountain J. Math. 14 119139. (1984)
+\bibitem[Goodloe 93]{GL93} Goodloe, A.; Loustaunau, P.
+``An abstract data type development of graded rings''
+In Fitch [Fit93], pp193202. ISBN 0387572724 (New York),
+3540572724 (Berlin). LCCN QA76.9.S88I576 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Conway 87]{CCNPW87} Conway, J.; Curtis, R.; Norton, S.; Parker, R.;
Wilson, R.
``Atlas of Finite Groups''
Oxford, Clarendon Press, 1987
+\bibitem[Gottliebsen 05]{GKM05} Gottliebsen, Hanne; Kelsey, Tom;
+Martin, Ursula
+``Hidden verification for computational mathematics''
+Journal of Symbolic Computation, Vol39, Num 5, pp539567 (2005)
+\verbwww.sciencedirect.com/science/article/pii/S0747717105000295
+%\verbaxiomdeveloper.org/axiomwebsite/papers/GKM05.pdf
+ keywords = "axiomref",
+ abstract = "
+ We present hidden verification as a means to make the power of
+ computational logic available to users of computer algebra systems
+ while shielding them from its complexity. We have implemented in PVS a
+ library of facts about elementary and transcendental function, and
+ automatic procedures to attempt proofs of continuity, convergence and
+ differentiability for functions in this class. These are called
+ directly from Maple by a simple pipelined interface. Hence we are
+ able to support the analysis of differential equations in Maple by
+ direct calls to PVS for: result refinement and verification, discharge
+ of verification conditions, harnesses to ensure more reliable
+ differential equation solvers, and verifiable lookup tables."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Conway 03]{CS03} Conway, John H.; Smith, Derek, A.
``On Quaternions and Octonions''
A.K Peters, Natick, MA. (2003) ISBN 1568811349
+\bibitem[Grabe 98]{Gra98} Gr\"abe, HansGert
+``About the Polynomial System Solve Facility of Axiom, Macyma, Maple
+Mathematica, MuPAD, and Reduce''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Gra98.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}
\begin{chunk}{ignore}
\bibitem[Cox 72]{Cox72} Cox M G
``The Numerical Evaluation of Bsplines''
J. Inst. Math. Appl. 10 134149. (1972)
+\bibitem[Grabmeier 91]{GHK91} Grabmeier, J.; Huber, K.; Krieger, U.
+``Das ComputeralgebraSystem AXIOM bei kryptologischen und
+verkehrstheoretischen Untersuchungen des
+Forschunginstituts der Deutschen Bundespost TELEKOM''
+Technischer Report TR 75.91.20, IBM Wissenschaftliches
+Zentrum, Heidelberg, Germany, 1991
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[CH 73]{CH73} Cox M G; Hayes J G
``Curve fitting: a guide and suite of algorithms for the
nonspecialist user''
Report NAC26. National Physical Laboratory. (1973)
+\bibitem[Grabmeier 92]{GS92} Grabmeier, J.; Scheerhorn, A.
+``Finite fields in Axiom''
+AXIOM Technical Report TR7/92 (ATR/5)(NP2522),
+Numerical Algorithms Group, Inc., Downer's
+Grove, IL, USA and Oxford, UK, 1992
+\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
+and Technical Report, IBM Heidelberg Scientific Center, 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cox 74a]{Cox74a} Cox M G
``A Datafitting Package for the Nonspecialist User''
Software for Numerical Mathematics. (ed D J Evans) Academic Press. (1974)
+\bibitem[Grabmeier 03]{GKW03} Grabmeier, Johannes; Kaltofen, Erich;
+Weispfenning, Volker (eds)
+Computer algebra handbook: foundations, applications, systems.
+SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+2003. ISBN 3540654666 637pp Includes CDROM
+\verbwww.springer.com/sgw/cda/frontpage/
+\verb0,11855,11022214778710,00.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cox 74b]{Cox74b} Cox M G
``Numerical methods for the interpolation and approximation of data
by spline functions''
PhD Thesis. City University, London. (1975)
+\bibitem[Griesmer 71]{GJ71} Griesmer, J. H.; Jenks, R.D.
+``SCRATCHPAD/1  an interactive facility for symbolic mathematics''
+In Petrick [Pet71], pp4258. LCCN QA76.5.S94 1971
+\verbdelivery.acm.org/10.1145/810000/806266/p42griesmer.pdf
+SYMSAC'71 Proc. second ACM Symposium on Symbolic and Algebraic
+Manipulation pp4548
+%\verbaxiomdeveloper.org/axiomwebsite/papers/GJ71.pdf REF:00027
+ 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}
\begin{chunk}{ignore}
\bibitem[Cox 75]{Cox75} Cox M G
``An Algorithm for Spline Interpolation''
J. Inst. Math. Appl. 15 95108. (1975)
+\bibitem[Griesmer 72a]{GJ72a} Griesmer, J.; Jenks, R.
+``Experience with an online symbolic math system SCRATCHPAD''
+in Online'72 [Onl72] ISBN 0903796023 LCCN QA76.55.O54 1972 Two volumes
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cox 77]{Cox77} Cox M G
``A Survey of Numerical Methods for Data and Function Approximation''
The State of the Art in Numerical Analysis. (ed D A H Jacobs)
Academic Press. 627668. (1977)
 keywords = "survey",
+\bibitem[Griesmer 72b]{GJ72b} Griesmer, James H.; Jenks, Richard D.
+``SCRATCHPAD: A capsule view''
+ACM SIGPLAN Notices, 7(10) pp93102, 1972. Proceedings of the symposium
+on Twodimensional manmachine communications. Mark B. Wells and
+James B. Morris (eds.).
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cox 78]{Cox78} Cox M G
``The Numerical Evaluation of a Spline from its Bspline Representation''
J. Inst. Math. Appl. 21 135143. (1978)
+\bibitem[Griesmer 75]{GJY75} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y
+``SCRATCHPAD User's Manual''
+IBM Research Publication RA70 June 1975
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Curtis 74]{CPR74} Curtis A R; Powell M J D; Reid J K
``On the Estimation of Sparse Jacobian Matrices''
J. Inst. Maths Applics. 13 117119. (1974)
+\bibitem[Griesmer 76]{GJY76} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y
+``A Set of SCRATCHPAD Examples''
+April 1976 (private copy)
+ keywords = "axiomref",
\end{chunk}
\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Dahlquist 74]{DB74} Dahlquist G; Bjork A
``Numerical Methods''
Prentice Hall. (1974)
+\bibitem[Gruntz 94]{GM94} Gruntz, D.; Monagan, M.
+``Introduction to Gauss''
+SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic
+Manipulation), 28(3) pp319 August 1994 CODEN SIGSBZ ISSN 01635824
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dalmas 98]{DA98} Dalmas, Stephane; Arsac, Olivier
``The INRIA OpenMath Library''
Projet SAFIR, INRIA Sophia Antipolis Nov 25, 1998
+\bibitem[Gruntz 96]{Gru96} Gruntz, Dominik
+``On Computing Limits in a Symbolic Manipulation System''
+Thesis, Swiss Federal Institute of Technology Z\"urich 1996
+Diss. ETH No. 11432
+\verbwww.cybertester.com/data/gruntz.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Gru96.pdf
+ keywords = "axiomref",
+ abstract = "
+ This thesis presents an algorithm for computing (onesided) limits
+ within a symbolic manipulation system. Computing limtis is an
+ important facility, as limits are used both by other functions such as
+ the definite integrator and to get directly some qualitative
+ information about a given function.
+
+ The algorithm we present is very compact, easy to understand and easy
+ to implement. It overcomes the cancellation problem other algorithms
+ suffer from. These goals were achieved using a uniform method, namely
+ by expanding the whole function into a series in terms of its most
+ rapidly varying subexpression instead of a recursive bottom up
+ expansion of the function. In the latter approach exact error terms
+ have to be kept with each approximation in order to resolve the
+ cancellation problem, and this may lead to an intermediate expression
+ swell. Our algorithm avoids this problem and is thus suited to be
+ implemented in a symbolic manipulation system."
\end{chunk}
+\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Dantzig 63]{Dan63} Dantzig G B
``Linear Programming and Extensions''
Princeton University Press. (1963)
+\bibitem[Boyle 88]{Boyl88} Boyle, Ann
+``Future Directions for Research in Symbolic Computation''
+Soc. for Industrial and Applied Mathematics, Philadelphia (1990)
+\verbwww.eecis.udel.edu/~caviness/wsreport.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/Boyl88.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport]{Dav} Davenport, James
``On Brillhart Irreducibility.''
To appear.
+\bibitem[Hassner 87]{HBW87} Hassner, Martin; Burge, William H.;
+Watt, Stephen M.
+``Construction of Algebraic Error Control Codes (ECC) on the Elliptic
+Riemann Surface''
+in [Wit87], pp58
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 93]{RefDav93} Davenport, J.H.
``Primality testing revisited''
Technical Report TR2/93
(ATR/6)(NP2556) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA
and Oxford, UK, August 1993
\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
+\bibitem[Heck 01]{Hec01} Heck, A.
+``Variables in computer algebra, mathematics and science''
+The International Journal of Computer Algebra in Mathematics Education
+Vol. 8 No. 3 pp195210 (2001)
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davis 67]{DR67} Davis P J; Rabinowitz P
``Numerical Integration''
Blaisdell Publishing Company. 3352. (1967)
+\bibitem[Huguet 89]{HP89} Huguet, L.; Poli, A. (eds).
+Applied Algebra, Algebraic Algorithms and ErrorCorrecting Codes.
+5th International Conference AAECC5 Proceedings.
+SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1989. ISBN 3540510826. LCCN QA268.A35 1987
+ keywords = "axiomref",
\end{chunk}
+\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Davis 75]{DR75} Davis P J; Rabinowitz P
``Methods of Numerical Integration''
Academic Press. (1975)
+\bibitem[Jacob 93]{JOS93} Jacob, G.; Oussous, N. E.; Steinberg, S. (eds)
+Proceedings SC 93
+International IMACS Symposium on Symbolic Computation. New Trends and
+Developments. LIFL Univ. Lille, Lille France, 1993
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[DeBoor 72]{DeB72} De Boor C
``On Calculating with Bsplines''
J. Approx. Theory. 6 5062. (1972)
+\bibitem[Janssen 88]{Jan88} Jan{\ss}en, R. (ed)
+Trends in Computer Algebra, International Symposium
+Bad Neuenahr, May 1921, 1987, Proceedings, volume 296 of Lecture Notes in
+Computer Science.
+SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1988 ISBN 3540189289, 0387189289 LCCN QA155.7.E4T74 1988
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[De Doncker 78]{DeD78} De Doncker E,
``An Adaptive Extrapolation Algorithm for Automatic Integration''
Signum Newsletter. 13 (2) 1218. (1978)
+\bibitem[Jenks 69]{Jen69} Jenks, R. D.
+``META/LISP: An interactive translator writing system''
+Research Report International Business Machines, Inc., Thomas J.
+Watson Research Center, Yorktown Heights, NY, USA, 1969 RC2968 July 1970
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Demmel 89]{Dem89} Demmel J W
``On Floatingpoint Errors in Cholesky''
LAPACK Working Note No. 14. University of Tennessee, Knoxville. 1989
+\bibitem[Jenks 71]{Jen71} Jenks, R. D.
+``META/PLUS: The syntax extension facility for SCRATCHPAD''
+Research Report RC 3259, International Business Machines, Inc., Thomas J.
+Watson Research Center, Yorktown Heights, NY, USA, 1971
+% REF:00040
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dennis 77]{DM77} Dennis J E Jr; More J J
``QuasiNewton Methods, Motivation and Theory''
SIAM Review. 19 4689. 1977
+\bibitem[Jenks 74]{Jen74} Jenks, R. D.
+``The SCRATCHPAD language''
+ACM SIGPLAN Notices, 9(4) pp101111 1974 CODEN SINODQ. ISSN 03621340
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dennis 81]{DS81} Dennis J E Jr; Schnabel R B
``A New Derivation of Symmetric PositiveDefinite Secant Updates''
Nonlinear Programming 4. (ed O L Mangasarian, R R Meyer and S M. Robinson)
Academic Press. 167199. (1981)
+\bibitem[Jen76]{Jen76} Jenks, Richard D.
+``A pattern compiler''
+In Richard D. Jenks, editor,
+SYMSAC '76: proceedings of the 1976 ACM Symposium on Symbolic and Algebraic
+Computation, August 1012, 1976, Yorktown Heights, New York, pp6065,
+ACM Press, New York, NY 10036, USA, 1976. LCCN QA155.7.EA .A15 1976
+QA9.58.A11 1976
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dennis 83]{DS83} Dennis J E Jr; Schnabel R B
``Numerical Methods for Unconstrained Optimixation and Nonlinear Equations''
PrenticeHall.(1983)
+\bibitem[Jenks 79]{Jen79} Jenks, R. D.
+``MODLISP: An Introduction''
+Proc EUROSAM 79, pp466480, 1979 and IBMRC8073 Jan 1980
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dierckx 75]{Die75} Dierckx P
``An Algorithm for Smoothing, Differentiating and Integration of
Experimental Data Using Spline Functions''
J. Comput. Appl. Math. 1 165184. (1975)
+\bibitem[Jenks 81]{JT81} Jenks, R.D.; Trager, B.M.
+``A Language for Computational Algebra''
+Proceedings of SYMSAC81, Symposium on Symbolic and Algebraic Manipulation,
+Snowbird, Utah August, 1981
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dierckx 81]{Die81} Dierckx P
``An Improved Algorithm for Curve Fitting with Spline Functions''
Report TW54. Dept. of Computer Science, Katholieke Universiteit Leuven. 1981
+\bibitem[Jenks 81a]{JT81a} Jenks, R.D.; Trager, B.M.
+``A Language for Computational Algebra''
+SIGPLAN Notices, New York: Association for Computing Machiner, Nov 1981
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dierckx 82]{Die82} Dierckx P
``A Fast Algorithm for Smoothing Data on a Rectangular Grid while using
Spline Functions''
SIAM J. Numer. Anal. 19 12861304. (1982)
+\bibitem[Jenks 81b]{JT81b} Jenks, R.D.; Trager, B.M.
+``A Language for Computational Algebra''
+IBM Research Report RC8930 IBM Yorktown Heights, NY
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dongarra 79]{DMBS79} Dongarra J J; Moler C B; Bunch J R;
Stewart G W
``LINPACK Users' Guide''
SIAM, Philadelphia. (1979)
+\bibitem[Jenks 84a]{Jen84a} Jenks, Richard D.
+``The new SCRATCHPAD language and system for computer algebra''
+In Golden and Hussain [GH84], pp409??
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dongarra 85]{DCHH85} Dongarra J J; Du Croz J J; Hammarling S;
Hanson R J
``A Proposal for an Extended set of Fortran Basic Linear
Algebra Subprograms''
SIGNUM Newsletter. 20 (1) 218. (1985)
+\bibitem[Jenks 84b]{Jen84b} Jenks, Richard D.
+``A primer: 11 keys to New Scratchpad''
+In Fitch [Fit84], pp123147. ISBN 038713350X LCCN QA155.7.E4 I57 1984
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dongarra 88]{REFDON88} Dongarra, Jack J.; Du Croz, Jeremy;
Hammarling, Sven; Hanson, Richard J.
``An Extended Set of FORTRAN Basic Linear Algebra Subroutines''
ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988,
pp 117
+\bibitem[Jenks 86]{JWS86} Jenks, Richard D.; Sutor, Robert S.;
+Watt, Stephen M.
+``Scratchpad II: An Abstract Datatype System for Mathematical Computation''
+Research Report RC 12327 (\#55257), Iinternational Business Machines, Inc.,
+Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1986 23pp
+\verbwww.csd.uwo.ca/~watt/pub/reprints/1987imaspadadt.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/JWS86.pdf
+ keywords = "axiomref",
+ abstract = "
+ Scratchpad II is an abstract datatype language and system that is
+ under development in the Computer Algebra Group, Mathematical Sciences
+ Department, at the IBM Thomas J. Watson Research Center. Some features
+ of APL that made computation particularly elegant have been borrowed.
+ Many different kinds of computational objects and data structures are
+ provided. Facilities for computation include symbolic integration,
+ differentiation, factorization, solution of equations and linear
+ algebra. Code economy and modularity is achieved by having
+ polymorphic packages of functions that may create datatypes. The use
+ of categories makes these facilities as general as possible."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dongarra 88a]{REFDON88a} Dongarra, Jack J.; Du Croz, Jeremy;
Hammarling, Sven; Hanson, Richard J.
``ALGORITHM 656: An Extended Set of Basic Linear Algebra Subprograms:
Model Implementation and Test Programs''
ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988,
pp 1832
+\bibitem[Jenks 87]{JWS87} Jenks, Richard D.; Sutor, Robert S.;
+Watt, Stephen M.
+``Scratchpad II: an Abstract Datatype System for Mathematical Computation''
+Proceedings Trends in Computer Algebra, Bad Neuenahr, LNCS 296,
+Springer Verlag, (1987)
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dongarra 90]{REFDON90} Dongarra, Jack J.; Du Croz, Jeremy;
Hammarling, Sven; Duff, Iain S.
``A Set of Level 3 Basic Linear Algebra Subprograms''
ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990,
pp 117
+\bibitem[Jenks 88]{JSW88} Jenks, R. D.; Sutor, R. S.; Watt, S. M.
+``Scratchpad II: An abstract datatype system for mathematical computation''
+In Jan{\ss}en [Jan88],
+pp12?? ISBN 3540189289, 0387189289 LCCN QA155.7.E4T74 1988
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dongarra 90a]{REFDON90a} Dongarra, Jack J.; Du Croz, Jeremy;
Hammarling, Sven; Duff, Iain S.
``ALGORITHM 679: A Set of Level 3 Basic Linear Algebra Subprograms:
Model Implementation and Test Programs''
ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990,
pp 1828
+\bibitem[Jenks 88a]{Jen88a} Jenks, R. D.
+``A Guide to Programming in BOOT''
+Computer Algebra Group, Mathematical Sciences Department, IBM Research
+Draft September 5, 1988
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ducos 00]{Duc00} Ducos, Lionel
``Optimizations of the subresultant algorithm''
Journal of Pure and Applied Algebra V145 No 2 Jan 2000 pp149163
+\bibitem[Jenks 88b]{Jen88b} Jenks, Richard
+``The Scratchpad II Computer Algebra System Interactive Environment Users
+Guide''
+ Spring 1988
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Duff 77]{Duff77} Duff I S,
``MA28  a set of Fortran subroutines for sparse unsymmetric linear
equations''
A.E.R.E. Report R.8730. HMSO. (1977)
+\bibitem[Jenks 88c]{JWS88} Jenks, R. D.; Sutor, R. S.; Watt, S. M.
+``Scratchpad II: an abstract datatype system for mathematical computation''
+In Jan{\ss}en
+[Jan88], pp1237. ISBN 3540189289, 0387189289 LCCN QA155.7.E4T74 1988
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Duval 96a]{Duva96a} Duval, D.; Gonz\'alezVega, L.
``Dynamic Evaluation and Real Closure''
Mathematics and Computers in Simulation 42 pp 551560 (1996)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva96a.pdf
+\begin{chunk}{axiom.bib}
+@book{Jenk92,
+ author = "Jenks, Richard D. and Sutor, Robert S.",
+ title = "AXIOM: The Scientific Computation System",
+ publisher = "SpringerVerlag, Berlin, Germany",
+ year = "1992",
+ isbn = "0387978550",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The aim of this paper is to present how the dynamic evaluation method
can be used to deal with the real closure of an ordered field. Two
kinds of questions, or tests, may be asked in an ordered field:
equality tests $(a=b?)$ and sign tests $(a > b?)$. Equality tests are
handled through splittings, exactly as in the algebraic closure of a
field. Sign tests are handled throug a structure called ``Tarski data
type''.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Duval 96]{Duva96} Duval, D.; Reynaud, J.C.
``Sketches and Computations over Fields''
Mathematics and Computers in Simulation 42 pp 363373 (1996)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva96.pdf
+\bibitem[Jenks 94]{JT94} Jenks, R. D.; Trager, B. M.
+``How to make AXIOM into a Scratchpad''
+In ACM [ACM94], pp3240 ISBN 0897916387 LCCN QA76.95.I59 1994
+%\verbaxiomdeveloper.org/axiomwebsite/papers/JT94.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The goal of this short paper is to describe one possible use of
sketches in computer algebra. We show that sketches are a powerful
tool for the description of mathematical structures and for the
description of computations.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Duval 94a]{Duva94a} Duval, D.; Reynaud, J.C.
``Sketches and Computation (Part I): Basic Definitions and Static Evaluation''
Mathematical Structures in Computer Science, 4, p 185238 Cambridge University Press (1994)
\verbjournals.cambridge.org/abstract_S0960129500000438
%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva94a.pdf
+\bibitem[Joswig 03]{JT03} Joswig, Michael; Takayama, Nobuki
+``Algebra, geometry, and software systems''
+SpringerVerlag ISBN 3540002561 p291
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We define a categorical framework, based on the notion of {\sl
sketch}, for specification and evaluation in the sense of algebraic
specifications and algebraic programming. This framework goes far
beyond our initial motivations, which was to specify computation with
algebraic numbers. We begin by redefining sketches in order to deal
explicitly with programs. Expressions and terms are carefully defined
and studied, then {\sl quasiprojective sketches} are introduced. We
describe {\sl static evaluation} in these sketches: we propose a
rigorous basis for evaluation in the corresponding structures. These
structures admit an initial model, but are not necessarily
equational. In Part II (Duval and Reynaud 1994), we study a more
general process, called {\sl dynamic evaluation}, for structures that
may have no initial model.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Duval 94b]{Duva94b} Duval, D.; Reynaud, J.C.
``Sketches and Computation (Part II): Dynamic Evaluation and Applications''
Mathematical Structures in Computer Science, 4, p 239271. Cambridge University Press (1994)
\verbjournals.cambridge.org/abstract_S096012950000044X
%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva94b.pdf
+\bibitem[Joyner 06]{J006} Joyner, David
+``OSCAS  Maxima''
+SIGSAM Communications in Computer Algebra, 157 2006
+\verbsage.math.washington.edu/home/wdj/sigsam/oscascca1.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In the first part of this paper (Duval and Reynaud 1994), we defined a
categorical framework, based on the notion of {\sl sketch}, for
specification and evaluation in the senses of algebraic specification
and algebraic programming. {\sl Static evaluation} in {\sl
quasiprojective sketches} was defined in Part I; in this paper, {\sl
dynamic evaluation} is introduced. It deals with more general
structures, which may have no initial model. Until now, this process
has not been used in algebraic specification systems, but computer
algebra systems are beginning to use it as a basic tool. Finally, we
give some applications of dynamic evaluation to computation in field
extensions.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Duval 94c]{Duva94c} Duval, Dominique
``Algebraic Numbers: An Example of Dynamic Evaluation''
J. Symbolic Computation 18, 429445 (1994)
\verbwww.sciencedirect.com/science/article/pii/S0747717106000551
%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva94c.pdf
+\bibitem[Joyner 14]{JO14} Joyner, David
+``Links to some open source mathematical programs''
+\verbwww.opensourcemath.org/opensource_math.html
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Dynamic evaluation is presented through examples: computations
involving algebraic numbers, automatic case discussion according to
the characteristic of a field. Implementation questions are addressed
too. Finally, branches are presented as ``dual'' to binary functions,
according to the approach of sketch theory.
\end{adjustwidth}

\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Fateman 08]{Fat08} Fateman, Richard
``Revisiting numeric/symbolic indefinite integration of rational functions, and extensions''
\verbwww.eecs.berkeley.edu/~fateman/papers/integ.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat08.pdf
+\bibitem[Kauers 08]{Kau08} Kauers, Manuel
+``Integration of Algebraic Functions: A Simple Heuristic for Finding
+the Logarithmic Part''
+ISSAC July 2008 ACM 978159593904 pp133140
+\verbwww.risc.jku.at/publications/download/risc_3427/Ka01.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Kau08.pdf
+ keywords = "axiomref",
+ abstract = "
+ A new method is proposed for finding the logarithmic part of an
+ integral over an algebraic function. The method uses Gr{\"o}bner bases
+ and is easy to implement. It does not have the feature of finding a
+ closed form of an integral whenever there is one. But it very often
+ does, as we will show by a comparison with the builtin integrators of
+ some computer algebra systems."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We know we can solve this problem: Given any rational function
$f(x)=p(x)/q(x)$, where $p$ and $q$ are univariate polynomials over
the rationals, compute its {\sl indefinite} integral, using if
necessary, algebraic numbers. But in many circumstances an approximate
result is more likely to be of use. Furthermore, it is plausible that
it would be more useful to solve the problem to allow definite
integration, or introduce additional parameters so that we can solve
multiple definite integrations. How can a computer algebra system
best answer the more useful questions? Finally, what if the integrand
is not a ratio of polynomials, but something more challenging?
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@misc{Flet01,
 author = "Fletcher, John P.",
 title = "Symbolic processing of Clifford Numbers in C++",
 year = "2001",
 journal = "Paper 25, AGACSE 2001."
}
+\begin{chunk}{ignore}
+\bibitem[Keady 94]{KN94} Keady, G.; Nolan, G.
+``Production of Argument SubPrograms in the AXIOM  NAG
+link: examples involving nonleanr systems''
+Technical Report TR1/94
+ATR/7 (NP2680), Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and
+Oxford, UK, 1994
+\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Flet09,
 author = "Fletcher, John P.",
 title = "Clifford Numbers and their inverses calculated using the matrix representation",
 publisher = "Chemical Engineering and Applied Chemistry, School of Engineering and Applied Science, Aston University, Aston Triangle, Birmingham B4 7 ET, U. K.",
 url = "http://www.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php"
}
+\begin{chunk}{ignore}
+\bibitem[Kelsey 99]{Kel99} Kelsey, Tom
+``Formal Methods and Computer Algebra: A Larch Specification of AXIOM
+Categories and Functors''
+Ph.D. Thesis, University of St Andrews, 1999
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fletcher 81]{Fle81} Fletcher R
``Practical Methods of Optimization''
Vol 2. Constrained Optimization. Wiley. (1981)
+\bibitem[Kelsey 00a]{Kel00a} Kelsey, Tom
+``Formal specification of computer algebra''
+University of St Andrews, 6th April 2000
+\verbwww.cs.standrews.cs.uk/~tom/pub/fscbs.ps
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Kel00a.pdf
+ keywords = "axiomref",
+ abstract = "
+ We investigate the use of formal methods languages and tools in the
+ design and development of computer algebra systems (henceforth CAS).
+ We demonstrate that errors in CAS design can be identified and
+ corrected by the use of (i) abstract specifications of types and
+ procedures, (ii) automated proofs of properties of the specifications,
+ and (iii) interface specifications which assist the verification of
+ pre and post conditions of implemented code."
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Floy63,
 author = "Floyd, R. W.",
 title = "Semantic Analysis and Operator Precedence",
 journal = "JACM",
 volume = "10",
 number = "3",
 pages = "316333",
 year = "1963"
}
+\begin{chunk}{ignore}
+\bibitem[Kelsey 00b]{Kel00b} Kelsey, Tom
+``Formal specification of computer algebra''
+(slides) University of St Andrews, Sept 21, 2000
+\verbwww.cs.standrews.cs.uk/~tom/pub/fscbstalk.ps
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Forsythe 57]{For57} Forsythe G E,
``Generation and use of orthogonal polynomials for data fitting
with a digital computer''
J. Soc. Indust. Appl. Math. 5 7488. (1957)
+\bibitem[Kendall 99a]{Ken99a} Kendall, W.S.
+``Itovsn3 in AXIOM: modules, algebras and stochastic differentials''
+\verbwww2.warwick.ac.uk/fac/sci/statistics/staff/academicresearch/
+\verbkendall/personal/ppt/328.ps.gz
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fortenbacher 90]{REFFor90} Fortenbacher, A.
``Efficient type inference and coercion in computer algebra''
Design and Implementation of Symbolic Computation Systems (DISCO 90)
A. Miola, (ed) vol 429 of Lecture Notes in Computer Science
SpringerVerlag, pp5660
+\bibitem[Kendall 99b]{Ken99b} Kendall, W.S.
+``Symbolic It\^o calculus in AXIOM: an ongoing story
+\verbwww2.warwick.ac.uk/fac/sci/statistics/staff/academicresearch/
+\verbkendall/personal/ppt/327.ps.gz
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Computer algebra systems of the new generation, like Scratchpad, are
characterized by a very rich type concept, which models the
relationship between mathematical domains of computation. To use these
systems interactively, however, the user should be freed of type
information. A type inference mechanism determines the appropriate
function to call. All known models which allow to define a semantics
for type inference cannot express the rich ``mathematical'' type
structure, so presently type inference is done heuristically. The
following paper defines a semantics for a subproblem thereof, namely
coercion, which is based on rewrite rules. From this definition, and
efficient coercion algorith for Scratchpad is constructed using graph
techniques.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Fox 68]{Fox68} Fox L.; Parker I B.
``Chebyshev Polynomials in Numerical Analysis''
Oxford University Press. (1968)
+\bibitem[Kosleff 91]{Kos91} P.V. Koseleff
+``Word games in free Lie algebras: several bases and formulas''
+Theoretical Computer Science 79(1) pp241256 Feb. 1991 CODEN TCSCDI
+ISSN 03043975
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Franke 80]{FN80} Franke R.; Nielson G
``Smooth Interpolation of Large Sets of Scattered Data''
Internat. J. Num. Methods Engrg. 15 16911704. (1980)
+\bibitem[Kusche 89]{KKM89} Kusche, K.; Kutzler, B.; Mayr, H.
+``Implementation of a geometry theorem proving package in SCRATCHPAD II''
+In Davenport [Dav89] pp246257 ISBN 3540515178 LCCN QA155.7.E4E86 1987
+ keywords = "axiomref",
\end{chunk}
+\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Fritsch 82]{Fri82} Fritsch F N
``PCHIP Final Specifications''
Report UCID30194. Lawrence Livermore National Laboratory. (1982)
+\bibitem[Lahey 08]{Lah08} Lahey, Tim
+``Sage Integration Testing''
+\verbgithub.com/tjl/sage_int_testing Dec. 2008
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fritsch 84]{FB84} Fritsch F N.; Butland J.
``A Method for Constructing Local Monotone Piecewise Cubic Interpolants''
SIAM J. Sci. Statist. Comput. 5 300304. (1984)
+\bibitem[Lambe 89]{Lam89} Lambe, L. A.
+``Scratchpad II as a tool for mathematical research''
+Notices of the AMS, February 1928 pp143147
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Froberg 65]{Fro65} Froberg C E.
``Introduction to Numerical Analysis''
AddisonWesley. 181187. (1965)
+\bibitem[Lambe 91]{Lam91} Lambe, L. A.
+``Resolutions via homological perturbation''
+Journal of Symbolic Computation 12(1) pp7187 July 1991
+CODEN JSYCEH ISSN 07477171
+ keywords = "axiomref",
\end{chunk}
\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Garcia 95]{Ga95} Garcia, A.; Stichtenoth, H.
``A tower of ArtinSchreier extensions of function fields attaining the
DrinfeldVladut bound''
Invent. Math., vol. 121, 1995, pp. 211222.
+\bibitem[Lambe 92]{Lam92} Lambe, Larry
+``Next Generation Computer Algebra Systems AXIOM and the Scratchpad
+Concept: Applications to Research in Algebra''
+$21^{st}$ Nordic Congress of Mathematicians 1992
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Lam92.pdf
+ keywords = "axiomref",
+ abstract = "
+ One way in which mathematicians deal with infinite amounts of data is
+ symbolic representation. A simple example is the quadratic equation
+ \[x = \frac{b\pm\sqrt{b^24ac}}{2a}\]
+ a formula which uses symbolic representation to describe the solutions
+ to an infinite class of equations. Most computer algebra systems can
+ deal with polynomials with symbolic coefficients, but what if symbolic
+ exponents are called for (e.g. $1+t^i$)? What if symbolic limits on
+ summations are also called for, for example
+ \[1+t+\ldots+t^i=\sum_j{t^j}\]
+
+ The ``Scratchpad Concept'' is a theoretical ideal which allows the
+ implementation of objects at this level of abstraction and beyond in a
+ mathematically consistent way. The Axiom computer algebra system is an
+ implementation of a major part of the Scratchpad Concept. Axiom
+ (formerly called Scratchpad) is a language with extensible
+ parameterized types and generic operators which is based on the
+ notions of domains and categories. By examining some aspects of the
+ Axiom system, the Scratchpad Concept will be illustrated. It will be
+ shown how some complex problems in homologicial algebra were solved
+ through the use of this system."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gathen 90a]{Gat90a} Gathen, Joachim von zur; Giesbrecht, Mark
``Constructing Normal Bases in Finite Fields''
J. Symbolic Computation pp 547570 (1990)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Gat90a.pdf
+\bibitem[Lambe 93]{Lam93} Lambe, Larry
+``On Using Axiom to Generate Code''
+(preprint) 1993
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
An efficient probabilistic algorithm to find a normal basis in a
finite field is presented. It can, in fact, find an element of
arbitrary prescribed additive order. It is based on a density estimate
for normal elements. A similar estimate yields a probabilistic
polynomialtime reduction from finding primitive normal elements to
finding primitive elements.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Gathen 90]{Gat90} Gathen, Joachim von zur
``Functional Decomposition Polynomials: the Tame Case''
Journal of Symbolic Computation (1990) 9, 281299
+\bibitem[Lambe 93a]{LL93} Lambe, Larry; Luczak, Richard
+``ObjectOriented Mathematical Programming and Symbolic/Numeric Interface''
+$3^{rd}$ International Conf. on Expert Systems in Numerical Computing 1993
+%\verbaxiomdeveloper.org/axiomwebsite/papers/LL93.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}
\begin{chunk}{axiom.bib}
@book{Gath99,
 author = {{von zur Gathen}, Joachim and Gerhard, J\"urgen},
 title = "Modern Computer Algebra",
 publisher = "Cambridge University Press",
 year = "1999",
 isbn = "0521641764"
}
+\begin{chunk}{ignore}
+\bibitem[Lebedev 08]{Leb08} Lebedev, Yuri
+``OpenMath Library for Computing on Riemann Surfaces''
+PhD thesis, Nov 2008 Florida State University
+\verbwww.math.fsu.edu/~ylebedev/research/HyperbolicGeometry.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gautschi 79a]{Gau79a} Gautschi W.
``A Computational Procedure for Incomplete Gamma Functions''
ACM Trans. Math. Softw. 5 466481. (1979)
+\bibitem[LeBlanc 91]{LeB91} LeBlanc, S.E.
+``The use of MathCAD and Theorist in the ChE classroom''
+In Anonymous [Ano91], pp287299 (vol. 1) 2 vols.
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gautschi 79b]{Gau79b} Gautschi W.
``Algorithm 542: Incomplete Gamma Functions''
ACM Trans. Math. Softw. 5 482489. (1979)
+\bibitem[Lecerf 96]{Le96} Lecerf, Gr\'egoire
+``Dynamic Evaluation and Real Closure Implementation in Axiom''
+June 29, 1996
+\verblecerf.perso.math.cnrs.fr/software/drc/drc.ps
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Le96.ps
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gentlemen 69]{Gen69} Gentlemen W M
``An Error Analysis of Goertzel's (Watt's) Method for Computing
Fourier Coefficients''
Comput. J. 12 160165. (1969)
+\bibitem[Lecerf 96a]{Le96a} Lecerf, Gr\'egoire
+``The Dynamic Real Closure implemented in Axiom''
+\verblecerf.perso.math.cnrs.fr/software/drc/drc.ps
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gentleman 73]{Gen73} Gentleman W M.
``Leastsquares Computations by Givens Transformations without Square Roots''
J. Inst. Math. Applic. 12 329336. (1973)
+\bibitem[Levelt 95]{Lev95} Levelt, A. H. M. (ed)
+ISSAC '95: Proceedings of the 1995 International
+Symposium on Symbolic and Algebraic Computation: July 1012, 1995, Montreal,
+Canada ISSACPROCEEDINGS1995. ACM Press, New York, NY 10036, USA, 1995
+ISBN 0897916999 LCCN QA76.95 I59 1995 ACM order number 505950
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gentleman 74]{Gen74} Gentleman W M.
``Algorithm AS 75. Basic Procedures for Large Sparse or
Weighted Linear Leastsquares Problems''
Appl. Statist. 23 448454. (1974)
+\bibitem[Li 06]{LM06} Li, Xin; Maza, Moreno
+``Efficient Implementation of Polynomial Arithmetic in a MultipleLevel
+Programming Environment''
+Lecture Notes in
+Computer Science Springer Vol 4151/2006 ISBN 9783540380849 pp1223
+Proceedings of International Congress of Mathematical Software ICMS 2006
+\verbwww.csd.uwo.ca/~moreno//Publications/LiMorenoMazaICMS06.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gentlemen 74a]{GM74a} Gentleman W. M.; Marovich S. B.
``More on algorithms that reveal properties of floating point
arithmetic units''
Comms. of the ACM, 17, 276277. (1974)
+\bibitem[Li 10]{YL10} Li, Yue; Dos Reis, Gabriel
+``A Quantitative Study of Reductions in Algebraic Libraries''
+PASCO 2010
+\verbwww.axiomatics.org/~gdr/concurrency/quantpasco10.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Genz 80]{GM80} Genz A C.; Malik A A.
``An Adaptive Algorithm for Numerical Integration over an Ndimensional
Rectangular Region''
J. Comput. Appl. Math. 6 295302. (1980)
+\bibitem[Li 11]{YL11} Li, Yue; Dos Reis, Gabriel
+``An Automatic Parallelization Framework for Algebraic Computation
+Systems''
+ISSAC 2011
+\verbwww.axiomatics.org/~gdr/concurrency/oaconcissac11.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/YL11.pdf
+ keywords = "axiomref",
+ abstract = "
+ This paper proposes a nonintrusive automatic parallelization
+ framework for typeful and propertyaware computer algebra systems."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 72]{GM72} Gill P E.; Miller G F.
``An Algorithm for the Integration of Unequally Spaced Data''
Comput. J. 15 8083. (1972)
+\bibitem[Ligatsikas 96]{Liga96} Ligatsikas, Zenon; Rioboo, Renaud;
+Roy, Marie Francoise
+``Generic computation of the real closure of an ordered field''
+Math. and Computers in Simulation 42 pp 541549 (1996)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Liga96.pdf
+ keywords = "axiomref",
+ abstract = "
+ This paper describes a generalization of the real closure computation
+ of an ordered field (Rioboo, 1991) enabling to use different technques
+ to code a single real algebraic number."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 74b]{GM74b} Gill P E.; Murray W. (eds)
``Numerical Methods for Constrained Optimization''
Academic Press. (1974)
+\bibitem[Linton 93]{Lin93} Linton, Steve
+``Vector Enumeration Programs, version 3.04''
+\verbwww.cs.standrews.ac.uk/~sal/nme/nme_toc.html#SEC1
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 76a]{GM76a} Gill P E.; Murray W.
``Minimization subject to bounds on the variables''
Report NAC 72. National Physical Laboratory. (1976)
+\bibitem[Liska 97]{LD97} Liska, Richard; Drska, Ladislav; Limpouch, Jiri;
+Sinor, Milan; Wester, Michael; Winkler, Franz
+``Computer Algebra  algorithms, systems and applications''
+June 2, 1997
+\verbkfe.fjfi.cvut.cz/~liska/ca/all.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 76b]{GM76b} Gill P E.; Murray W.
``Algorithms for the Solution of the Nonlinear Leastsquares Problem''
NAC 71 National Physical Laboratory. (1976)
+\bibitem[Lucks 86]{Luc86} Lucks, Michael
+``A fast implementation of polynomial factorization''
+In Bruce W. Char, editor, Proceedings of the 1986 Symposium on Symbolic
+and Algebraic Computation: SYMSAC '86, July 2123, 1986, Waterloo, Ontario,
+pp228232 ACM Press, New York, NY 10036, USA, 1986. ISBN 0897911997
+LCCN QA155.7.E4 A281 1986 ACM order number 505860
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 78]{GM78} Gill P E.; Murray W.
``Algorithms for the Solution of the Nonlinear Leastsquares Problem''
SIAM J. Numer. Anal. 15 977992. (1978)
+\bibitem[Lueken 77]{Lue77} Lueken, E.
+``Ueberlegungen zur Implementierung eines Formelmanipulationssystems''
+Master's thesis, Technischen Universit{\"{a}}t CaroloWilhelmina zu
+Braunschweig. Braunschweig, Germany, 1977
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 79]{GM79} Gill P E.; Murray W;
``Conjugategradient Methods for Largescale Nonlinear Optimization''
Technical Report SOL 7915. Department of Operations Research,
Stanford University. (1979)
+\bibitem[Lynch 91]{LM91} Lynch, R.; Mavromatis, H. A.
+``New quantum mechanical perturbation technique
+using an 'electronic scratchpad' on an inexpensive computer''
+American Journal of Pyhsics, 59(3) pp270273, March 1991.
+CODEN AJPIAS ISSN 00029505
+ keywords = "axiomref",
\end{chunk}
+\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Gill 81]{GMW81} Gill P E.; Murray W.; Wright M H.
``Practical Optimization''
Academic Press. 1981
+\bibitem[Mahboubi 05]{Mah05} Mahboubi, Assia
+``Programming and certifying the CAD algorithm inside the coq system''
+Mathematics, Algorithms, Proofs, volume 05021 of Dagstuhl
+Seminar Proceedings, Schloss Dagstuhl (2005)
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 82]{GMW82} Gill P E.; Murray W.; Saunders M A.; Wright M H.
``The design and implementation of a quadratic programming algorithm''
Report SOL 827. Department of Operations Research,
Stanford University. (1982)
+\bibitem[Mathews 89]{Mat89} Mathews, J.
+``Symbolic computational algebra applied to Picard iteration''
+Mathematics and computer education, 23(2) pp117122 Spring 1989 CODEN MCEDDA,
+ISSN 07308639
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 84a]{GMSW84a} Gill P E.; Murray W.; Saunders M A.; Wright M H
``User's Guide for SOL/QPSOL Version 3.2''
Report SOL 845. Department of Operations Research, Stanford University. 1984
+\bibitem[McJones 11]{McJ11} McJones, Paul
+``Software Presentation Group  Common Lisp family''
+\verbwww.softwarepreservation.org/projects/LISP/common_lisp_family
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 84b]{GMSW84b} Gill P E.; Murray W.; Saunders M A.; Wright M H
``Procedures for Optimization Problems with a Mixture of
Bounds and General Linear Constraints''
ACM Trans. Math. Softw. 10 282298. 1984
+\bibitem[Melachrinoudis 90]{MR90} Melachrinoudis, E.; Rumpf, D. L.
+``Teaching advantages of transparent computer software  MathCAD''
+CoED, 10(1) pp7176, JanuaryMarch 1990 CODEN CWLJDP ISSN 07368607
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 86a]{GMSW86a} Gill P E.; Hammarling S.; Murray W.;
Saunders M A.; Wright M H.
``User's Guide for LSSOL (Version 1.0)''
Report SOL 861. Department of Operations Research, Stanford University. 1986
+\bibitem[Miola 90]{Mio90} Miola, A. (ed)
+``Design and Implementation of Symbolic Computation Systems''
+International Symposium DISCO '90, Capri, Italy, April 1012, 1990, Proceedings
+volume 429 of Lecture Notes in Cmputer Science,
+SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1990 ISBN 0387525319 (New York), 3540525319 (Berlin) LCCN QA76.9.S88I576
+1990
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 86b]{GMSW86b} Gill P E.; Murray W.; Saunders M A.; Wright M H.
``Some Theoretical Properties of an Augmented Lagrangian Merit Function''
Report SOL 866R. Department of Operations Research, Stanford University. 1986
+\bibitem[Miola 93]{Mio93} Miola, A. (ed)
+``Design and Implementation of Symbolic Computation Systems''
+International Symposium DISCO '93 Gmunden, Austria, September 1517, 1993:
+Proceedings.
+SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1993 ISBN 354057235X LCCN QA76.9.S88I576 1993
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gladwell 79]{Gla79} Gladwell I
``Initial Value Routines in the NAG Library''
ACM Trans Math Softw. 5 386400. (1979)
+\bibitem[Missura 94]{Miss94} Missura, Stephan A.; Weber, Andreas
+``Using Commutativity Properties for Controlling Coercions''
+\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/
+\verbWeberA/MissuraWeber94a.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Miss94.pdf
+ keywords = "axiomref",
+ abstract = "
+ This paper investigates some soundness conditions which have to be
+ fulfilled in systems with coercions and generic operators. A result of
+ Reynolds on unrestricted generic operators is extended to generic
+ operators which obey certain constraints. We get natural conditions
+ for such operators, which are expressed within the theoretic framework
+ of category theory. However, in the context of computer algebra, there
+ arise examples of coercions and generic operators which do not fulfil
+ these conditions. We describe a framework  relaxing the above
+ conditions  that allows distinguishing between cases of ambiguities
+ which can be resolved in a quite natural sense and those which
+ cannot. An algorithm is presented that detects such unresolvable
+ ambiguities in expressions."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gladwell 80]{GS80} Gladwell I.; Sayers D K
``Computational Techniques for Ordinary Differential Equations''
Academic Press. 1980
+\bibitem[Monagan 87]{Mon87} Monagan, Michael B.
+``Support for Data Structures in Scratchpad II''
+in [Wit87], pp1718
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gladwell 86]{Gla86} Gladwell I
``Vectorisation of one dimensional quadrature codes''
Techincal Report. TR7/86 NAG. (1986)
+\bibitem[Monagan 93]{Mon93} Monagan, M. B.
+``Gauss: a parameterized domain of computation system with
+support for signature functions''
+In Miola [Mio93], pp8194 ISBN 354057235X LCCN QA76.9.S88I576 1993
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gladwell 87]{Gla87} Gladwell I
``The NAG Library Boundary Value Codes''
Numerical Analysis Report. 134 Manchester University. (1987)
+\bibitem[Mora 89]{Mor89} Mora, T. (ed)
+Applied Algebra, Algebraic Algorithms and ErrorCorrecting
+Codes, 6th International Conference, AAECC6, Rome, Italy, July 48, 1998,
+Proceedings, volume 357 of Lecture Notes in Computer Science
+SpringerVerlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1989 ISBN 3540510834, LCCN QA268.A35 1988 Conference held jointly with
+ISSAC '88
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Goedel 40]{God40} Goedel
``The consistency of the continuum hypothesis''
Ann. Math. Studies, Princeton Univ. Press, 1940
+\bibitem[Moses 71]{Mos71} Moses, Joel
+``Algebraic Simplification: A Guide for the Perplexed''
+CACM August 1971 Vol 14 No. 8 pp527537
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Goldman 87]{Gold87} Goldman, L.
``Integrals of multinomial systems of ordinary differential equations''
J. of Pure and Applied Algebra, 45, 225240 (1987)
\verbwww.sciencedirect.com/science/article/pii/0022404987900727/pdf
\verb?md5=5a0c70643eab514ccf47d80e4fc6ec5a&
\verbpid=1s2.00022404987900727main.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Gold87.pdf
+\bibitem[Moses 08]{Mos08} Moses, Joel
+``Macsyma: A Personal History''
+Invited Presentation in Milestones in Computer Algebra, May 2008, Tobago
+\verbesd.mit.edu/Faculty_Pages/moses/Macsyma.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Mos08.pdf
+ keywords = "axiomref",
+ abstract = "
+ The Macsyma system arose out of research on mathematical software in
+ the AI group at MIT in the 1960's. Algorithm development in symbolic
+ integration and simplification arose out of the interest of people,
+ such as the author, who were also mathematics students. The later
+ development of algorithms for the GCD of sparse polynomials, for
+ example, arose out of the needs of our user community. During various
+ times in the 1970's the computer on which Macsyma ran was one of the
+ most popular notes on the ARPANET. We discuss the attempts in the late
+ 70's and the 80's to develop Macsyma systems that ran on popular
+ computer architectures. Finally, we discuss the impact of the
+ fundamental ideas in Macsyma on current research on large scale
+ engineering systems."
\end{chunk}
+\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Gollan 90]{GG90} H. Gollan; J. Grabmeier
``Algorithms in Representation Theory and
their Realization in the Computer Algebra System Scratchpad''
Bayreuther Mathematische Schriften, Heft 33, 1990, 123
+\bibitem[Naylor]{NPxx} Naylor, William; Padget, Julian
+``From Untyped to Polymorphically Typed Objects in Mathematical Web
+Services''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/NPxx.pdf
+ 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}
\begin{chunk}{ignore}
\bibitem[Golub 89]{GL89} Golub, Gene H.; Van Loan, Charles F.
``Matrix Computations''
Johns Hopkins University Press ISBN 0801837723 (1989)
+\bibitem[Naylor 95]{N95} Naylor, Bill
+``Symbolic Interface for an advanced hyperbolic PDE solver''
+\verbwww.sci.csd.uwo.ca/~bill/Papers/symbInterface2.ps
+%\verbaxiomdeveloper.org/axiomwebsite/papers/N95.pdf
+ keywords = "axiomref",
+ abstract = "
+ An Axiom front end is described, which is used to generate
+ mathematical objects needed by one of the latest NAG routines, to be
+ included in the Mark 17 version of the NAG Numerical library. This
+ routine uses powerful techniques to find the solution to Hyperbolic
+ Partial Differential Equations in conservation form and in one spatial
+ dimension. These mathematical objects are nontrivial, requiring much
+ mathematical knowledge on the part of the user, which is otherwise
+ irrelvant to the physical problem which is to be solved. We discuss
+ the individual mathematical objects, considering the mathematical
+ theory which is relevant, and some of the problems which have been
+ encountered and solved during the FORTRAN generation necessary to
+ realise the object. Finally we display some of our results."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Golub 96]{GL96} Golub, Gene H.; Van Loan, Charles F.
``Matrix Computations''
Johns Hopkins University Press ISBN 9780801854149 (1996)
+\bibitem[Naylor 00b]{ND00} Naylor, W.A.; Davenport, J.H.
+``A MonteCarlo Extension to a CategoryBased Type System''
+\verbwww.sci.csd.uwo.ca/~bill/Papers/monteCarCat3.ps
+%\verbaxiomdeveloper.org/axiomwebsite/papers/ND00.pdf
+ keywords = "axiomref",
+ abstract = "
+ The normal claim for mathematics is that all calculations are 100\%
+ accurate and therefore one calculation can rely completely on the
+ results of subcalculations, hoever there exist {\sl MonteCarlo}
+ algorithms which are often much faster than the equivalent
+ deterministic ones where the results will have a prescribed
+ probability (presumably small) of being incorrect. However there has
+ been little discussion of how such algorithms can be used as building
+ blocks in Computer Algebra. In this paper we describe how the
+ computational category theory which is the basis of the type structure
+ used in the Axiom computer algebra system may be extended to cover
+ probabilistic algorithms, which use MonteCarlo techniques. We follow
+ this with a specific example which uses Straight Line Program
+ representation."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Grabmeier]{Grab} Grabmeier, J.
``On Plesken's root finding algorithm''
in preparation
+\bibitem[Norman 75]{Nor75} Norman, A. C.
+``Computing with formal power series''
+ACM Transactions on Mathematical Software, 1(4) pp346356
+Dec. 1975 CODEN ACMSCU ISSN 00983500
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Grebmeier 87]{GK87} Grabmeier, J.; Kerber, A.;
``The Evaluation of Irreducible Polynomial Representations of the General
Linear Groups and of the Unitary Groups over Fields of Characteristic 0''
Acta Appl. Math. 8 (1987), 271291
+\bibitem[Norman 75a]{Nor75a} Norman, A.C.
+``The SCRATCHPAD Power Series Package''
+IBM T.J. Watson Research RC4998
+ keywords = "axiomref",
\end{chunk}
+\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Grabmeier 92]{REFGS92} Grabmeier, J.; Scheerhorn, A.
``Finite fields in Axiom''
AXIOM Technical Report TR7/92 (ATR/5)(NP2522),
Numerical Algorithms Group, Inc., Downer's
Grove, IL, USA and Oxford, UK, 1992
\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
+\bibitem[Ollivier 89]{Oll89} Ollivier, F.
+``Inversibility of rational mappings and structural
+identifiablility in automatics''
+In ACM [ACM89], pp4354 ISBN 0897913256 LCCN QA76.95.I59 1989
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Granville 1911]{Gran1911} Granville, William Anthony
``Elements of the Differential and Integral Calculus''
\verbdjm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Gran1911.pdf
+\bibitem[Online 72]{Onl72}.
+Online 72: conference proceedings ... international conference on online
+interactive computing, Brunel University, Uxbridge, England, 47 September
+1972 ISBN 0903796023 LCCN QA76.55.O54 1972 Two volumes.
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gruntz 93]{Gru93} Gruntz, Dominik
``Limit computation in computer algebra''
\verbalgo.inria.fr/seminars/sem9293/gruntz.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Gru93.pdf
+\bibitem[OpenMath]{OpenMa}.
+``OpenMath Technical Overview''
+\verbwww.openmath.org/overview/technical.html
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The automatic computation of limits can be reduced to two main
subproblems. The first one is asymptotic comparison where one must
decide automatically which one of two functions in a specified class
dominates the other one asymptotically. The second one is asymptotic
cancellation and is usually exemplified by
\[e^x[exp(1/x+e^{x})exp(1/x)],\quad{}x \leftarrow \infty\]

In this example, if the sum is expanded in powers of $1/x$, the
expansion always yields $O(x^{k})$, and this is not enough to
conclude.
+\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In 1990, J.Shackell found an algorithm that solved both these problems
for the case of $explog$ functions, i.e. functions build by recursive
application of exponential, logarithm, algebraic extension and field
operations to one variable and the rational numbers. D. Gruntz and
G. Gonnet propose a slightly different algorithm for explog
functions. Extensions to larger classes of functions are also
discussed.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Page 07]{Pa07} Page, William S.
+``Axiom  Open Source Computer Algebra System''
+Poster ISSAC 2007 Proceedings Vol 41 No 3 Sept 2007 p114
+ keywords = "axiomref",
\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{axiom.bib}
@article{Hach95,
 author = "Hach\'e, G. and Le Brigand, D.",
 title = "Effective construction of algebraic geometry codes",
 journal = "IEEE Transaction on Information Theory",
 volume = "41",
 month = "November",
 year = "1995",
 pages = "16151628"
}
+\begin{chunk}{ignore}
+\bibitem[Petitot 90]{Pet90} Petitot, Michel
+``Types r\'ecursifs en scratchpad, application aux polyn\^omes non
+commutatifs''
+LIFL, 1990
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Hach95a,
 author = "Hach\'e, G.",
 title = "Computation in algebraic function fields for effective construction of algebraicgeometric codes",
 journal = "Lecture Notes in Computer Science",
 volume = "948",
 year = "1995",
 pages = "262278"
}
+\begin{chunk}{ignore}
+\bibitem[Petitot 93]{Pet93} Petitot, M.
+``Experience with Axiom''
+In Jacob et al. [JOS93], page 240
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@phdthesis{Hach96,
 author = "Hach\'e, G.",
 title = "Construction effective des codes g\'eom\'etriques",
 school = "l'Universit\'e Pierre et Marie Curie (Paris 6)",
 year = "1996",
 month = "Septembre"
}
+\begin{chunk}{ignore}
+\bibitem[Petric 71]{Pet71} Petric, S. R. (ed)
+Proceedings of the second symposium on Symbolic and
+Algebraic Manipulation, March 2325, 1971, Los Angeles, California, ACM Press,
+New York, NY 10036, USA, 1971. LCCN QA76.5.S94 1971
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Hall 76]{HW76} Hall G.; Watt J M. (eds),
``Modern Numerical Methods for Ordinary Differential Equations''
Clarendon Press. (1976)
+\bibitem[Pinch 93]{Pin93} Pinch, R.G.E.
+``Some Primality Testing Algorithms''
+Devlin, Keith (ed.)
+Computers and Mathematics November 1993, Vol 40, Number 9 pp12031210
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Hamdy 04]{Ham04} Hamdy, S.
``LiDIA A library for computational number theory''
Reference manual Edition 2.1.1 May 2004
\verbwww.cdc.informatik.tudarmstadt.de/TI/LiDIA
+\bibitem[Poll (b)]{Polxx} Poll, Erik
+``The type system of Axiom''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Polxx.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Hammarling 85]{Ham85} Hammarling S.
`` The Singular Value Decomposition in Multivariate Statistics''
ACM Signum Newsletter. 20, 3 225. (1985)
+\bibitem[Purtilo 86]{Pur86} Purtilo, J.
+``Applications of a software interconnection system in mathematical
+problem solving environments'' In Bruce W. Char, editor. Proceedings of the
+1986 Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 2123,
+ACM Press, New York, NY 10036, USA, 1986. ISBN 0897911997 LCCN QA155.7.E4
+A281 1986 ACM order number 505860
+ keywords = "axiomref",
\end{chunk}
+\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Hammersley 67]{HH67} Hammersley J M; Handscomb D C.
``MonteCarlo Methods''
Methuen. (1967)
+\bibitem[Rainer 14]{Rain14} Joswig, Rainer
+``2014: 30+ Years Common Lisp the Language''
+\verblispm.de/30ycltl
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Hath1896,
 author = "Hathway, Arthur S.",
 title = "A Primer Of Quaternions",
 year = "1896"
}
+\begin{chunk}{ignore}
+\bibitem[Rioboo 03a]{Riob03a} Rioboo, Renaud
+``Quelques aspects du calcul exact avec des nombres r\'eels''
+Ph.D. Thesis, Laboratoire d'Informatique Th\'eorique et Programmationg
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Riob03a.ps
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Haya05,
 author = "Hayashi, K. and Kangkook, J. and Lascu, O. and Pienaar, H. and Schreitmueller, S. and Tarquinio, T. and Thompson, J.",
 title = "AIX 5L Practical Performance Tools and Tuning Guide",
 publisher = "IBM",
 year = "2005",
 url = "http://www.redbooks.ibm.com/redbooks/pdfs/sg246478.pdf",
 paper = "Haya05.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Rioboo 03]{Riob03} Rioboo, Renaud
+``Towards Faster Real Algebraic Numbers''
+J. of Symbolic Computation 36 pp 513533 (2003)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Riob03.pdf
+ keywords = "axiomref",
+ abstract = "
+ This paper presents a new encoding scheme for real algebraic number
+ manipulations which enhances current Axiom's real closure. Algebraic
+ manipulations are performed using different instantiations of
+ subresultantlike algorithms instead of Euclideanlike algorithms.
+ We use these algorithms to compute polynomial gcds and Bezout
+ relations, to compute the roots and the signs of algebraic
+ numbers. This allows us to work in the ring of real algebraic integers
+ instead of the field of read algebraic numbers avoiding many
+ denominators."
\end{chunk}
+
\begin{chunk}{ignore}
\bibitem[Hayes 70]{Hay70} Hayes J G.
``Curve Fitting by Polynomials in One Variable''
Numerical Approximation to Functions and Data.
(ed J G Hayes) Athlone Press, London. (1970)
+\bibitem[Robidoux 93]{Rob93} Robidoux, Nicolas
+``Does Axiom Solve Systems of O.D.E's Like Mathematica?''
+July 1993
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Rob93.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}
\begin{chunk}{ignore}
\bibitem[Hayes 74]{Hay74} Hayes J G.
``Numerical Methods for Curve and Surface Fitting''
Bull Inst Math Appl. 10 144152. (1974)
+\bibitem[Rioboo 92]{Rio92} Rioboo, R.
+``Real algebraic closure of an ordered field, implementation in Axiom''
+In Wang [Wan92], pp206215 ISBN 0897914899 (soft cover)
+0897914902 (hard cover) LCCN QA76.95.I59 1992
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Rio92.pdf
+ keywords = "axiomref",
+ 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
+ roots of a polynomial. This process is designed to be recursively
+ used, so the resulting domain of computation is the set of all real
+ algebraic numbers. An implementation for the real algebraic closure
+ has been done in Axiom (previously called Scratchpad)."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Hayes 74a]{HH74} Hayes J G.; Halliday J,
``The Leastsquares Fitting of Cubic Spline Surfaces to General Data Sets''
J. Inst. Math. Appl. 14 89103. (1974)
+\bibitem[Roesner 95]{Roe95} Roesner, K. G.
+``Verified solutions for parameters of an exact solution for
+nonNewtonian liquids using computer algebra'' Zeitschrift fur Angewandte
+Mathematik und Physik, 75 (suppl. 2):S435S438, 1995 ISSN 00442267
+ keywords = "axiomref",
\end{chunk}
+\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Henrici 56]{Hen56} Henrici, Peter
``Automatic Computations with Power Series''
Journal of the Association for Computing Machinery, Volume 3, No. 1,
January 1956, 1015
+\bibitem[Sage 14]{Sage14} Stein, William
+``Sage''
+\verbwww.sagemath.org/doc/reference/interfaces/sage/interfaces/axiom.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Higham 88]{Hig88} Higham, N.J.
``FORTRAN codes for estimating the onenorm of a
real or complex matrix, with applications to condition estimation''
ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381396, December 1988.
+\bibitem[Salvy 89]{Sal89} Salvy, B.
+``Examples of automatic asymptotic expansions''
+Technical Report 114,
+Inst. Nat. Recherche Inf. Autom., Le Chesnay, France, Dec. 1989 18pp
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Higham 02]{Hig02} Higham, Nicholas J.
``Accuracy and stability of numerical algorithms''
SIAM Philadelphia, PA ISBN 0898715210 (2002)
+\bibitem[Salvy 91]{Sal91} Salvy, B.
+``Examples of automatic asymptotic expansions''
+SIGSAM Bulletin (ACM Special Interest Group on Symbolic and
+Algebraic Manipulation), 25(2) pp417
+April 1991 CODEN SIGSBZ ISSN 01635824
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Hock 81]{HS81} Hock W.; Schittkowski K.
``Test Examples for Nonlinear Programming Codes''
Lecture Notes in Economics and Mathematical Systems. 187 SpringerVerlag. 1981
+\begin{chunk}{axiom.bib}
+@article{Saun80,
+ author = "Saunders, B. David",
+ title = "A Survey of Available Systems",
+ journal = "SIGSAM Bull.",
+ issue_date = "November 1980",
+ volume = "14",
+ number = "4",
+ month = "November",
+ year = "1980",
+ issn = "01635824",
+ pages = "1228",
+ numpages = "17",
+ url = "http://doi.acm.org/10.1145/1089235.1089237",
+ doi = "10.1145/1089235.1089237",
+ acmid = "1089237",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ keywords = "axiomref,survey",
+ paper = "Saun80.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Householder 70]{Hou70} Householder A S.
``The Numerical Treatment of a Single Nonlinear Equation''
McGrawHill. (1970)
+\bibitem[Schu 92]{Sch92} Sch\"u, J.
+``Implementing des CartanKuranishiTheorems in AXIOM''
+Master's diploma thesis (in german), Institut f\"ur Algorithmen und
+Kognitive Systeme, Universit\"t Karlsruhe 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Hous81,
 author = "Householder, Alston S.",
 title = "Principles of Numerical Analysis",
 publisher = "Dover Publications, Mineola, NY",
 year = "1981",
 isbn = "048645312X"
}
+\begin{chunk}{ignore}
+\bibitem[Schwarz 88]{Sch88} Schwarz, F.
+``Programming with abstract data types: the symmetry package SPDE
+in Scratchpad''
+In Jan{\ss}en [Jan88], pp167176, ISBN 3540189289,
+0387189289 LCCN QA155.7.E4T74 1988
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Huang 96]{HI96} Huang, M.D.; Ierardi, D.
``Efficient algorithms for RiemannRoch problem and for addition in the
jacobian of a curve''
Proceedings 32nd Annual Symposium on Foundations of Computer Sciences.
IEEE Comput. Soc. Press, pp. 678687.
+\bibitem[Schwarz 89]{Sch89} Schwarz, F.
+``A factorization algorithm for linear ordinary differential equations''
+In ACM [ACM89], pp1725 ISBN 0897913256 LCCN QA76.95.I59 1989
+ keywords = "axiomref",
\end{chunk}
\subsection{I} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{ignore}
+\bibitem[Schwarz 91]{Sch91} Schwarz, F.
+``Monomial orderings and Gr{\"o}bner bases''
+SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic
+Manipulation) 2591) pp1023 Jan. 1991 CODEN SIGSBZ ISSN 01635824
+ keywords = "axiomref",
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[IBM]{IBM}.
SCRIPT Mathematical Formula Formatter User's Guide, SH206453,
IBM Corporation, Publishing Systems Information Development,
Dept. G68, P.O. Box 1900, Boulder, Colorado, USA 803019191.
+\bibitem[Seiler 94]{Sei94} Seiler, Werner Markus
+``Analysis and Application of the Formal Theory of Partial Differential
+Equations''
+PhD thesis, School of Physics and Materials, Lancaster University (1994)
+\verbwww.mathematik.unikassel.de/~seiler/Papers/Diss/diss.ps.gz
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Sei94.pdf
+ keywords = "axiomref",
+ abstract = "
+ An introduction to the formal theory of partial differential equations
+ is given emphasizing the properties of involutive symbols and
+ equations. An algorithm to complete any differential equation to an
+ involutive one is presented. For an involutive equation possible
+ values for the number of arbitrary functions in its general solution
+ are determined. The existence and uniqueness of solutions for analytic
+ equations is proven. Applications of these results include an
+ analysis of symmetry and reduction methods and a study of gauge
+ systems. It is show that the Dirac algorithm for systems with
+ constraints is closely related to the completion of the equation of
+ motion to an involutive equation. Specific examples treated comprise
+ the YangMills Equations, Einstein Equations, complete and Jacobian
+ systems, and some special models in two and three dimensions. To
+ facilitate the involved tedious computations an environment for
+ geometric approaches to differential equations has been developed in
+ the computer algebra system Axiom. The appendices contain among others
+ brief introductions into CartenK{\"a}hler Theory and JanetRiquier
+ Theory."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Itoh 88]{Itoh88} Itoh, T.;, Tsujii, S.
``A fast algorithm for computing multiplicative inverses
in $GF(2^m)$ using normal bases''
Inf. and Comp. 78, pp.171177, 1988
%\verbaxiomdeveloper.org/axiomwebsite/Itoh88.pdf
+\bibitem[Seiler 94a]{Sei94a} Seiler, W.M.
+``Completion to involution in AXIOM''
+in Calmet [Cal94] pp103104
+ keywords = "axiomref",
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper proposes a fast algorithm for computing multiplicative
inverses in $GF(2^m)$ using normal bases. Normal bases have the
following useful property: In the case that an element $x$ in
$GF(2^m)$ is represented by normal bases, $2^k$ power operation of an
element $x$ in $GF(2^m)$ can be carried out by $k$ times cyclic shift
of its vector representation. C.C. Wang et al. proposed an algorithm
for computing multiplicative inverses using normal bases, which
requires $(m2)$ multiplications in $GF(2^m)$ and $(m1)$ cyclic
shifts. The fast algorithm proposed in this paper also uses normal
bases, and computes multiplicative inverses iterating multiplications
in $GF(2^m)$. It requires at most $2[log_2(m1)]$ multiplications in
$GF(2^m)$ and $(m1)$ cyclic shifts, which are much less than those
required in Wang's method. The same idea of the proposed fast
algorithm is applicable to the general power operation in $GF(2^m)$
and the computation of multiplicative inverses in $GF(q^m)$ $(q=2^n)$.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Iyanaga 77]{Iya77} Iyanaga, Shokichi; Iyanaga, Yukiyosi Kawada
``Encyclopedic Dictionary of Mathematics''
1977
+\bibitem[Sieler 94b]{Sei94b} Seiler, W.M.
+``Pseudo differential operators and integrable systems in AXIOM''
+Computer Physics Communications, 79(2) pp329340 April 1994 CODEN CPHCBZ
+ISSN 00104655
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Sei94b.pdf
+ keywords = "axiomref",
+ 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}
\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Jacobson 68]{Jac68} Jacobson, N.
``Structure and Representations of Jordan Algebras''
AMS, Colloquium Publications Volume 39
+\bibitem[Seiler 95]{Sei95} Seiler, W.M.
+``Applying AXIOM to partial differential equations''
+Internal Report 9517, Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik
+1995
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Sei95.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}
\begin{chunk}{ignore}
\bibitem[James 81]{JK81} James, Gordon; Kerber, Adalbert
``The Representation Theory of the Symmetric Group''
Encyclopedia of Mathematics and its Applications Vol. 16
AddisonWesley, 1981
+\bibitem[Seiler 95b]{SC95} Seiler, W.M.; Calmet, J.
+``JET  An Axiom Environment for Geometric Computations with Differential
+Equations''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/SC95.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}
\begin{chunk}{ignore}
\bibitem[Jaswon 77]{JS77} Jaswon, M A.; Symm G T.
``Integral Equation Methods in Potential Theory and Elastostatics''
Academic Press. (1977)
+\bibitem[Seiler 97]{Sei97} Seiler, Werner M.
+``Computer Algebra and Differential Equations: An Overview''
+\verbwww.mathematik.unikassel.di/~seiler/Papers/Postscript/CADERep.ps.gz
+ keywords = "axiomref",
+ abstract = "
+ We present an informal overview of a number of approaches to
+ differential equations which are popular in computer algebra. This
+ includes symmetry and completion theory, local analysis, differential
+ ideal and Galois theory, dynamical systems and numerical analysis. A
+ large bibliography is provided."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jeffrey 04]{Je04} Jeffrey, Alan
``Handbook of Mathematical Formulas and Integrals''
Third Edition, Elsevier Academic Press ISBN 0123822564
+\bibitem[Seiler (a)]{Seixx} Seiler, W.M.
+``DETools: A Library for Differential Equations''
+\verbiakswww.ira.uka.de/iakscalmet/werner/werner.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenning 66]{Jen66} Jennings A
``A Compact Storage Scheme for the Solution of Symmetric Linear
Simultaneous Equations''
Comput. J. 9 281285. (1966)
+\bibitem[Shannon 88]{SS88} Shannon, D.; Sweedler, M.
+``Using Gr{\"o}bner bases to determine algebra
+membership, split surjective algebra homomorphisms determine birational
+equivalence''
+Journal of Symbolic Computation 6(23) pp267273
+Oct.Dec. 1988 CODEN JSYCEH ISSN 07477171
+ keywords = "axiomref",
\end{chunk}
\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Kalkbrener 91]{Kal91} Kalkbrener, M.
``Three contributions to elimination theory''
Ph. D. Thesis, University of Linz, Austria, 1991
+\bibitem[Sit 89]{Sit89} Sit, W.Y.
+``On Goldman's algorithm for solving firstorder multinomial
+autonomous systems'' In Mora [Mor89], pp386395 ISBN 3540510834
+LCCN QA268.A35 1998 Conference held jointly with ISSAC '88
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Kalkbrener 98]{Kal98} Kalkbrener, M.
``Algorithmic properties of polynomial rings''
Journal of Symbolic Computation 1998
+\bibitem[Sit 92]{Sit92} Sit, W.Y.
+``An algorithm for solving parametric linear systems''
+Journal of Symbolic Computations, 13(4) pp353394, April 1992 CODEN JSYCEH
+ISSN 07477171
+\verbwww.sciencedirect.com/science/article/pii/S0747717108801046/pdf
+\verb?md5=00aa65e18e6ea5c4a008c8dfdfcd4b83&
+\verbpid=1s2.0S0747717108801046main.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Sit92.pdf
+ keywords = "axiomref",
+ abstract = "
+ We present a theoretical foundation for studying parametric systesm of
+ linear equations and prove an efficient algorithm for identifying all
+ parametric values (including degnerate cases) for which the system is
+ consistent. The algorithm gives a small set of regimes where for each
+ regime, the solutions of the specialized systems may be given
+ uniformly. For homogeneous linear systems, or for systems were the
+ right hand side is arbitrary, this small set is irredunant. We discuss
+ in detail practical issues concerning implementations, with particular
+ emphasis on simplification of results. Examples are given based on a
+ close implementation of the algorithm in SCRATCHPAD II. We also give a
+ complexity analysis of the Gaussian elimination method and compare
+ that with our algorithm."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Kantor 89]{Kan89} Kantor,I.L.; Solodovnikov, A.S.
``Hypercomplex Numbers''
Springer Verlag Heidelberg, 1989, ISBN 0387969802
+\bibitem[Sit 06]{Sit06} Sit, Emil
+``Tools for Repeatable Research''
+\verbwww.emilsit.net/blog/archives/toolsforrepeatableresearch
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Kaufmann 00]{KMJ00} Kaufmann, Matt; Manolios, Panagiotis;
Moore J Strother
``ComputerAided Reasoning: An Approach''
Springer, July 31. 2000 ISBN 0792377443
+\bibitem[Smedley 92]{Sme92} Smedley, Trevor J.
+``Using pictorial and object oriented programming for computer algebra''
+In Hal Berghel et al., editors. Applied computing 
+technologicial challenges of the 199s: proceedings of the 1992 ACM/SIGAPP
+Symposium on Applied Computing, Kansas City Convention Center, March 13, 1992
+pp12431247. ACM Press, New York, NY 10036, USA, 1992. ISBN 089791502X
+LCCN QA76.76.A65 S95 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Knuth 71]{Knu71} Knuth, Donald
``The Art of Computer Programming''
2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing,
AddisonWesley 1971, p. 397398
+\bibitem[Smith 07]{SDJ07} Smith, Jacob; Dos Reis, Gabriel; Jarvi, Jaakko
+``Algorithmic differentiation in Axiom''
+ACM SIGSAM ISSAC Proceedings 2007 Waterloo, Canada 2007 pp347354
+ISBN 9781595937438
+%\verbaxiomdeveloper.org/axiomwebsite/papers/SDJ07.pdf
+ keywords = "axiomref",
+ abstract = "
+ This paper describes the design and implementation of an algorithmic
+ differentiation framework in the Axiom computer algebra system. Our
+ implementation works by transformations on Spad programs at the level
+ of the typed abstract syntax tree."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Knuth 84]{Knu84} Knuth, Donald
{\it The \TeX{}book}.
Reading, Massachusetts, AddisonWesley Publishing Company, Inc.,
1984. ISBN 0201134489
+\bibitem[SSC92]{SSC92}.
+``Algorithmic Methods For Lie Pseudogroups''
+In N. Ibragimov, M. Torrisi and A. Valenti, editors, Proc. Modern Group
+Analysis: Advanced Analytical and Computational Methods in Mathematical
+Physics, pp337344, Acireale (Italy), 1992 Kluwer, Dordrecht 1993
+\verbiakswww.ira.uka.de/iakscalmet/werner/Papers/Acireale92.ps.gz
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Knut92,
 author = "Knuth, Donald E.",
 title = "Literate Programming",
 publisher = "Center for the Study of Language and Information, Stanford CA",
 year = "1992",
 isbn = "0937073814"
}
+\begin{chunk}{ignore}
+\bibitem[SSV87]{SSV87} Senechaud, P.; Siebert, F.; Villard G.
+``Scratchpad II: Pr{\'e}sentation d'un nouveau langage de calcul formel''
+Technical Report 640M, TIM 3 (IMAG), Grenoble, France, Feb 1987
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Knu98]{Knu98} Donald Knuth
``The Art of Computer Programming'' Vol. 3
(Sorting and Searching)
AddisonWesley 1998
+\bibitem[Steele]{Steele} Steele, Guy L.; Gabriel, Richard P.
+``The Evolution of Lisp''
+\verbwww.dreamsongs.com/Files/HOPL2Uncut.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Kobayashi 89]{Koba89} Kobayashi, H.; Moritsugu, S.; Hogan, R.W.
``On Radical ZeroDimensional Ideals''
J. Symbolic Computations 8, 545552 (1989)
\verbwww.sciencedirect.com/science/article/pii/S0747717189800604/pdf
\verb?md5=f06dc6269514c90dcae57f0184bcbe65&
\verbpid=1s2.0S0747717189800604main.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Koba88.pdf
+\bibitem[Sutor 85]{Sut85} Sutor, R.S.
+``The Scratchpad II computer algebra language and system''
+In Buchberger and Caviness [BC85], pp3233 ISBN 0387159835 (vol. 1),
+0387159843 (vol. 2) LCCN QA155.7.E4 E86 1985 Two volumes.
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Kolchin 73]{Kol73} Kolchin, E.R.
``Differential Algebra and Algebraic Groups''
(Academic Press, 1973).
+\bibitem[Sutor 87a]{SJ87a} Sutor, R. S.; Jenks, R. D.
+``The type inference and coercion facilities in
+the Scratchpad II interpreter'' In Wexelblat [Wex87], pp5663
+ISBN 0897912357 LCCN QA76.7.S54 v22:7 SIGPLAN Notices, v22 n7 (July 1987)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/SJ87a.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Koutschan 10]{Kou10} Koutschan, Christoph
``Axiom / FriCAS''
\verbwww.risc.jku.at/education/courses/ws2010/cas/axiom.pdf
+\bibitem[Sutor 87b]{Su87} Sutor, Robert S.
+``The Scratchpad II Computer Algebra System. Using and
+Programming the Interpreter''
+IBM Course presentation slide deck Spring 1987
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Kozen 86]{KL86} Kozen, Dexter; Landau, Susan
``Polynomial Decomposition Algorithms''
Journal of Symbolic Computation (1989) 7, 445456
+\bibitem[Sutor 87c]{SJ87c} Sutor, Robert S.; Jenks, Richard
+``The type inference and coercion facilities
+in the Scratchpad II interpreter''
+Research report RC 12595 (\#56575),
+IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1987, 11pp
+%\verbaxiomdeveloper.org/axiomwebsite/papers/SJ87c.pdf
+ keywords = "axiomref",
+ abstract = "
+ The Scratchpad II system is an abstract datatype programming language,
+ a compiler for the language, a library of packages of polymorphic
+ functions and parameterized abstract datatypes, and an interpreter
+ that provides sophisticated type inference and coercion facilities.
+ Although originally designed for the implementation of symbolic
+ mathematical algorithms, Scratchpad II is a general purpose
+ programming language. This paper discusses aspects of the
+ implementation of the intepreter and how it attempts to provide a user
+ friendly and relatively weakly typed front end for the strongly typed
+ programming language."
\end{chunk}
\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{ignore}
+\bibitem[Sutor 88]{Su88} Sutor, Robert S.
+``A guide to programming in the scratchpad 2 interpreter''
+IBM Manual, March 1988
+ keywords = "axiomref",
+\end{chunk}
\begin{chunk}{axiom.bib}
@book{Lamp86,
 author = "Lamport, Leslie",
 title = "LaTeX: A Document Preparation System",
 publisher = "AddisonWesley Publishing Company, Reading, Massachusetts",
 year = "1986",
 isbn = "020115790X"
}
+\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Thompson 00]{Tho00} Thompson, Simon
+``Logic and dependent types in the Aldor Computer Algebra System''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Tho00.pdf
+ keywords = "axiomref",
+ 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
+ type system with {\sl type evaluation} reported elsewhere. The logic
+ is used to provide an axiomatisation of a number of familiar Aldor
+ categories as well as a type of vectors."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lautrup 71]{Lau71} Lautrup B.
``An Adaptive Multidimensional Integration Procedure''
Proc. 2nd Coll. on Advanced Methods in Theoretical Physics, Marseille. (1971)
+\bibitem[Thompson (a)]{TTxx} Thompson, Simon; Timochouk, Leonid
+``The Aldor\\ language''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/TTxx.pdf
+ keywords = "axiomref",
+ abstract = "
+ This paper introduces the \verbAldor language, which is a
+ functional programming language with dependent types and a powerful,
+ typebased, overloading mechanism. The language is built on a subset
+ of Aldor, the 'library compiler' language for the Axiom computer
+ algebra system. \verbAldor is designed with the intention of
+ incorporating logical reasoning into computer algebra computations.
+
+ The paper contains a formal account of the semantics and type system
+ of \verbAldor; a general discussion of overloading and how the
+ overloading in \verbAldor fits into the general scheme; examples
+ of logic within \verbAldor and notes on the implementation of the
+ system."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lawson 77]{Law77} Lawson C L.
``Software for C Surface Interpolation''
Mathematical Software III. (ed J R Rice) Academic Press. 161194. (1977)
+\bibitem[Touratier 98]{Tou98} Touratier, Emmanuel
+``Etude du typage dans le syst\`eme de calcul scientifique Aldor''
+Universit\'e de Limoges 1998
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Tou98.pdf
+ keywords = "axiomref",
\end{chunk}
+\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Lawson 74]{LH74} Lawson C L.; Hanson R J.
``Solving Leastsquares Problems''
PrenticeHall. (1974)
+\bibitem[van der Hoeven 14]{JvdH14} van der Hoeven, Joris
+``Computer algebra systems and TeXmacs''
+\verbwww.texmacs.org/tmweb/plugins/cas.en.html
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Laws79,
 author = "Lawson, C.L. and Hanson R.J. and Kincaid, D.R. and Krogh, F.T.",
 title = "Algorithm 539: Basic linear algebra subprograms for FORTRAN usage",
 journal = "ACM Transactions on Mathematical Software",
 volume = "5",
 number = "3",
 month = "September",
 year = "1979",
 pages = "308323"
+@article{Hoei94,
+ author = "{van Hoeij}, M.",
+ title = "An algorithm for computing an integral basis in an algebraic
+ function field",
+ journal = "Journal of Symbolic Computation",
+ volume = "18",
+ number = "4",
+ year = "1994",
+ pages = "353363",
+ issn = "07477171",
+ keywords = "axiomref",
+ paper = "Hoei94.pdf",
+ abstract = "
+ Algorithms for computing integral bases of an algebraic function field
+ are implemented in some computer algebra systems. They are used e.g.
+ for the integration of algebraic functions. The method used by Maple
+ 5.2 and AXIOM is given by Trager in [Trag84]. He adapted an algorithm
+ of Ford and Zassenhaus [Ford, 1978], that computes the ring of
+ integers in an algebraic number field, to the case of a function field.
+
+ It turns out that using algebraic geometry one can write a faster
+ algorithm. The method we will give is based on Puiseux expansions.
+ One cas see this as a variant on the Coates' algorithm as it is
+ described in [Davenport, 1981]. Some difficulties in computing with
+ Puiseux expansions can be avoided using a sharp bound for the number
+ of terms required which will be given in Section 3. In Section 5 we
+ derive which denominator is needed in the integral basis. Using this
+ result 'intermediate expression swell' can be avoided.
+
+ The Puiseux expansions generally introduce algebraic extensions. These
+ extensions will not appear in the resulting integral basis."
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lawson 79]{LHKK79} Lawson C L; Hanson R J; Kincaid D R;
 Krogh F T
``Basic Linear Algebra Subprograms for Fortran Usage''
ACM Trans. Math. Softw. 5 308325. (1979)
+\begin{chunk}{axiom.bib}
+@misc{Hoei08,
+ author = "{van Hoeij}, Mark and Novocin, Andrew",
+ title = "A Reduction Algorithm for Algebraic Function Fields",
+ year = "2008",
+ month = "April",
+ url = "http://andy.novocin.com/pro/algext.pdf",
+ paper = "Hoei08.pdf",
+ abstract = "
+ Computer algebra systesm often produce large expressions involving
+ complicated algebraic numbers. In this paper we study variations of
+ the {\tt polred} algorithm that can often be used to find better
+ representations for algebraic numbers. The main new algorithm
+ presented here is an algorithm that treats the same problem for the
+ function field case."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lazard 91]{Laz91} Lazard, D.
``A new method for solving algebraic systems of positive dimension''
Discr. App. Math. 33:147160,1991
+\bibitem[Vasconcelos 99]{Vas99} Vasconcelos, Wolmer
+``Computational Methods in Commutative Algebra and Algebraic Geometry''
+Springer, Algorithms and Computation in Mathematics, Vol 2 1999
+ISBN 3540213112
+ keywords = "axiomref",
\end{chunk}
+\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Lazard92]{Laz92} Lazard, D.
``Solving Zerodimensional Algebraic Systems''
Journal of Symbolic Computation, 1992, 13, 117131
+\bibitem[Wang 89]{Wan89} Wang, D.
+``A program for computing the Liapunov functions and Liapunov
+constants in Scratchpad II''
+SIGSAM Bulletin (ACM Special Interest Group
+on Symbolic and Algebraic Manipulation), 23(4) pp2531, Oct. 1989,
+CODEN SIGSBZ ISSN 01635824
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Laza90,
 author = "Lazard, Daniel and Rioboo, Renaud",
 title = "Integration of rational functions: Rational computation of the logarithmic part",
 journal = "Journal of Symbolic Computation",
 volume = "9",
 number = "2",
 year = "1990",
 month = "February",
 pages = "113115",
+\begin{chunk}{ignore}
+\bibitem[Wang 91]{Wan91} Wang, Dongming
+``Mechanical manipulation for a class of differential systems''
+Journal of Symbolic Computation, 12(2) pp233254 Aug. 1991
+CODEN JSYCEH ISSN 07477171
keywords = "axiomref",
 paper = "Laza90.pdf"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A new formula is given for the logarithmic part of the integral of a
rational function, one that strongly improves previous algorithms and
does not need any computation in an algebraic extension of the field
of constants, nor any factorisation since only polynomial arithmetic
and GCD computations are used. This formula was independently found
and implemented in SCRATCHPAD by B.M. Trager.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{LeBr88,
 author = "Le Brigand, D.; Risler, J.J.",
 title = "Algorithme de BrillNoether et codes de Goppa",
 journal = "Bull. Soc. Math. France",
 volume = "116",
 year = "1988",
 pages = "231253"
}
+\begin{chunk}{ignore}
+\bibitem[Wang 92]{Wan92} Wang, Paul S. (ed)
+International System Symposium on Symbolic and
+Algebraic Computation 92 ACM Press, New York, NY 10036, USA, 1992
+ISBN 0897914899 (soft cover), 0897914902 (hard cover),
+LCCN QA76.95.I59 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Lege11,
 author = "Legendre, George L. and Grazini, Stefano",
 title = "Pasta by Design",
 publisher = "Thames and Hudson",
 isbn = "9780500515808",
 year = "2011"
}
+\begin{chunk}{ignore}
+\bibitem[Watanabe 90]{WN90} Watanabe, Shunro; Nagata, Morio; (ed)
+ISSAC '90 Proceedings of the
+International Symposium on Symbolic and Algebraic Computation ACM Press,
+New York, NY, 10036, USA. 1990 ISBN 0897914015 LCCN QA76.95.I57 1990
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lenstra 87]{LS87} Lenstra, H. W.; Schoof, R. J.
``Primitivive Normal Bases for Finite Fields''
Math. Comp. 48, 1987, pp. 217231
+\bibitem[Watt 85]{Wat85} Watt, Stephen
+``Bounded Parallelism in Computer Algebra''
+PhD Thesis, University of Waterloo
+\verbwww.csd.uwo.ca/~watt/pub/reprints/1985smwphd.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Leop03,
 author = "Leopardi, Paul",
 title = "A quick introduction to Clifford Algebras",
 publisher = "School of Mathematics, University of New South Wales",
 year = "2003",
 paper = "Leop03.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Watt 86]{Wat86} Watt, S.M.; Della Dora, J.
+``Algebra Snapshot: Linear Ordinary Differential Operators''
+Scratchpad II Newsletter: Vol 1 Num 2 (Jan 1986)
+\verbwww.csd.uwo.ca/~watt/pub/reprints/1986snewslodo.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lewis 77]{Lew77} Lewis J G,
``Algorithms for sparse matrix eigenvalue problems''
Technical Report STANCS77595. Computer Science Department,
Stanford University. (1977)
+\bibitem[Watt 87]{Wat87} Watt, Stephen
+``Domains and Subdomains in Scratchpad II''
+in [Wit87], pp35
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lidl 83]{LN83} Lidl, R.; Niederreiter, H.
``Finite Field, Encycoldia of Mathematics and Its Applications''
Vol. 20, Cambridge Univ. Press, 1983 ISBN 0521302404
+\bibitem[Watt 87a]{WB87} Watt, Stephen M.; Burge, William H.
+``Mapping as First Class Objects''
+in [Wit87], pp1317
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Linger 79]{LMW79} Linger, Richard C.; Mills, Harlan D.;
Witt, Bernard I.
``Structured Programming: Theory and Practice''
AddisonWesley (March 1979) ISBN 0201144611
+\bibitem[Watt 89]{Wat89} Watt, S. M.
+``A fixed point method for power series computation''
+In Gianni [Gia89], pp206217 ISBN 3540510842 LCCN QA76.95.I57
+1988 Conference held jointly with AAECC6
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lipson 81]{Lip81} Lipson, D.
``Elements of Algebra and Algebraic Computing''
The Benjamin/Cummings Publishing Company, Inc.Menlo Park, California, 1981.
+\bibitem[Watt 90]{WJST90} Watt, S.M.; Jenks, R.D.; Sutor, R.S.; Trager B.M.
+``The Scratchpad II type system: Domains and subdomains''
+in A.M. Miola, editor Computing Tools
+for Scientific Problem Solving, Academic Press, New York, 1990
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Loet09,
 author = "Loetzsch, Martin and Bleys, Joris and Wellens, Pieter",
 title = "Understanding the Dynamics of Complex Lisp Programs",
 year = "2009",
 url = "http://www.martinloetzsch.de/papers/loetzsch09understanding.pdf",
 paper = "Loet09.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Watt 91]{Wat91} Watt, Stephen M. (ed)
+Proceedings of the 1991 International Symposium on
+Symbolic and Algebraic Computation, ISSAC'91, July 1517, 1991, Bonn, Germany,
+ACM Press, New York, NY 10036, USA, 1991 ISBN 0897914376
+LCCN QA76.95.I59 1991
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Loet00,
 author = "Loetzsch, M.",
 title = "GTFL  A graphical terminal for Lisp",
 year = "2000",
 url = "http://martinloetzsch.de/gtfl"
}
+\begin{chunk}{ignore}
+\bibitem[Watt 94a]{Wat94a} Watt, Stephen M.; Dooley, S.S.; Morrison, S.C.;
+Steinback, J.M.; Sutor, R.S.
+``A\# User's Guide''
+Version 1.0.0 O($\epsilon{}^1$) June 8, 1994
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Losc60,
 author = {L\"osch, Friedrich},
 title = "Tables of Higher Functions",
 publisher = "McGrawHill Book Company",
 year = "1960"
}
+\begin{chunk}{ignore}
+\bibitem[Watt 94b]{Wat94} Watt, Stephen M.; Broadbery, Peter A.;
+Dooley, Samuel S.; Iglio, Pietro
+``A First Report on the A\# Compiler (including benchmarks)''
+IBM Research Report RC19529 (85075) May 12, 1994
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Wat94.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[LTU10]{LTU10}.
``Lambda the Ultimate''
\verblambdatheultimate.org/node/3663#comment62440
+\bibitem[Watt 94c]{Wat94c} Watt, Stephen M.
+``A\# Language Reference Version 0.35''
+IBM Research Division Technical Report RC19530 May 1994
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Luke69a,
 author = "Luke, Yudell L.",
 title = "The Special Functions and their Approximations",
 volume = "1",
 publisher = "Academic Press",
 year = "1969",
 booktitle = "Mathematics in Science and Engineering Volume 53I"
}
+\begin{chunk}{ignore}
+\bibitem[Watt 95]{Wat95} Watt, S.M.; Broadbery, P.A.; Dooley, S.S.; Iglio, P.
+Steinbach, J.M.; Morrison, S.C.; Sutor, R.S.
+``AXIOM Library Compiler Users Guide''
+The Numerical Algorithms Group (NAG) Ltd, 1994
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Luke69b,
 author = "Luke, Yudell L.",
 title = "The Special Functions and their Approximations",
 volume = "2",
 publisher = "Academic Press",
 year = "1969",
 booktitle = "Mathematics in Science and Engineering Volume 53I"
}
+\begin{chunk}{ignore}
+\bibitem[Watt 01]{Wat01} Watt, Stephen M.; Broadbery, Peter A.; Iglio, Pietro;
+Morrison, Scott C.; Steinbach, Jonathan M.
+``FOAM: A First Order Abstract Machine Version 0.35''
+IBM T. J. Watson Research Center (2001)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Wat01.pdf
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lyness 83]{Lyn83} Lyness J N.
``When not to use an automatic quadrature routine''
SIAM Review. 25 6387. (1983)
+\bibitem[Weber 92]{Webe92} Weber, Andreas
+``Type Systems for Computer Algebra''
+\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber92a.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe92.pdf
+ keywords = "axiomref",
+ abstract = "
+ An important feature of modern computer algebra systems is the support
+ of a rich type system with the possibility of type inference. Basic
+ features of such a type system are polymorphism and coercion between
+ types. Recently the use of ordersorted rewrite systems was proposed
+ as a general framework. We will give a quite simple example of a
+ family of types arising in computer algebra whose coercion relations
+ cannot be captured by a finite set of firstorder rewrite rules."
\end{chunk}
\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Mac Lane 79]{MB79} Mac Lane, Saunders; Birkhoff, Garret
``Algebra''
AMS Chelsea Publishing ISBN 0821816462
+\bibitem[Weber 92b]{Webe92b} Weber, Andreas
+``Structuring the Type System of a Computer Algebra System''
+\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber92a.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe92b.pdf
+ keywords = "axiomref",
+ abstract = "
+ Most existing computer algebra systems are pure symbol manipulating
+ systems without language support for the occuring types. This is
+ mainly due to the fact taht the occurring types are much more
+ complicated than in traditional programming languages. In the last
+ decade the study of type systems has become an active area of
+ research. We will give a proposal for a type system showing that
+ several problems for a type system of a symbolic computation system
+ can be solved by using results of this research. We will also provide
+ a variety of examples which will show some of the problems that remain
+ and that will require further research."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Malcolm 72]{Mal72} Malcolm M. A.
``Algorithms to reveal properties of floatingpoint arithmetic''
Comms. of the ACM, 15, 949951. (1972)
+\bibitem[Weber 93b]{Webe93b} Weber, Andreas
+``Type Systems for Computer Algebra''
+\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber93b.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe93b.pdf
+ keywords = "axiomref",
+ abstract = "
+ We study type systems for computer algebra systems, which frequently
+ correspond to the ``pragmatically developed'' typing constructs used
+ in AXIOM. A central concept is that of {\sl type classes} which
+ correspond to AXIOM categories. We will show that types can be
+ syntactically described as terms of a regular ordersorted signature
+ if no type parameters are allowed. Using results obtained for the
+ functional programming language Haskell we will show that the problem
+ of {\sl type inference} is decidable. This result still holds if
+ higherorder functions are present and {\sl parametric polymorphism}
+ is used. These additional typing constructs are useful for further
+ extensions of existing computer algebra systems: These typing concepts
+ can be used to implement category theoretic constructs and there are
+ many well known constructive interactions between category theory and
+ algebra."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Malcolm 76]{MS76} Malcolm M A.; Simpson R B.
``Local Versus Global Strategies for Adaptive Quadrature''
ACM Trans. Math. Softw. 1 129146. (1976)
+\bibitem[Weber 94]{Web94} Weber, Andreas
+``Algorithms for Type Inference with Coercions''
+ISSAC 94 ACM 0897916387/94/0007
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Web94.pdf
+ keywords = "axiomref",
+ abstract = "
+ This paper presents algorithms that perform a type inference for a
+ type system occurring in the context of computer algebra. The type
+ system permits various classes of coercions between types and the
+ algorithms are complete for the precisely defined system, which can be
+ seen as a formal description of an important subset of the type system
+ supported by the computer algebra program Axiom.
+
+ Previously only algorithms for much more restricted cases of coercions
+ have been described or the frameworks used have been so general that
+ the corresponding type inference problems were known to be
+ undecidable."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Marden 66]{Mar66} Marden M.
``Geometry of Polynomials''
Mathematical Surveys. 3 Am. Math. Soc., Providence, RI. (1966)
+\bibitem[Weber 95]{Webe95} Weber, A.
+``On coherence in computer algebra''
+\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber94e.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe95.pdf
+ keywords = "axiomref",
+ abstract = "
+ Modern computer algebra systems (e.g. AXIOM) support a rich type
+ system including parameterized data types and the possibility of
+ implicit coercions between types. In such a type system it will be
+ frequently the case that there are different ways of building
+ coercions between types. An important requirement is that all
+ coercions between two types coincide, a property which is called {\sl
+ coherence}. We will prove a coherence theorem for a formal type system
+ having several possibilities of coercions covering many important
+ examples. Moreover, we will give some informal reasoning why the
+ formally defined restrictions can be satisfied by an actual system."
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Mars07,
 author = "Marshak, U.",
 title = "HTAJAX  AJAX framework for Hunchentoot",
 year = "2007",
 url = "http://commonlisp.net/project/htajax/htajax.html"
}
+\begin{chunk}{ignore}
+\bibitem[Weber 96]{Webe96} Weber, Andreas
+``Computing Radical Expressions for Roots of Unity''
+\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber96a.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe96.pdf
+ keywords = "axiomref",
+ abstract = "
+ We present an improvement of an algorithm given by Gauss to compute a
+ radical expression for a $p$th root of unity. The time complexity of
+ the algorithm is $O(p^3m^6log p)$, where $m$ is the largest prime
+ factor of $p1$."
\end{chunk}
+
\begin{chunk}{ignore}
\bibitem[Maza 95]{MR95} Maza, M. Moreno; Rioboo, R.
``Computations of gcd over algebraic towers of simple extensions''
In proceedings of AAECC11 Paris, 1995.
+\bibitem[Weber 99]{Webe99} Weber, Andreas
+``Solving Cyclotomic Polynomials by Radical Expressions''
+\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/
+\verbWeberA/WeberKeckeisen99a.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe99.pdf
+ keywords = "axiomref",
+ abstract = "
+ We describe a Maple package that allows the solution of cyclotomic
+ polynomials by radical expressions. We provide a function that is an
+ extension of the Maple {\sl solve} command. The major algorithmic
+ ingredient of the package is an improvement of a method due to Gauss
+ which gives radical expressions for roots of unity. We will give a
+ summary for computations up to degree 100, which could be done within
+ a few hours of cpu time on a standard workstation."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Maza 97]{Maz97} Maza, M. Moreno
``Calculs de pgcd audessus des tours
d'extensions simples et resolution des systemes d'equations algebriques''
These, Universite P.etM. Curie, Paris, 1997.
+\bibitem[WeiJiang 12]{WJ12} WeiJiang
+``Top free algebra System''
+\verbweijiang.com/it/software/topfreealgebrasystembyemathematicabyemaple
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Maza 98]{Maz98} Maza, M. Moreno
``A new algorithm for computing triangular
decomposition of algebraic varieties''
 NAG Tech. Rep. 4/98.
+\bibitem[Wester 99]{Wes99} Wester, Michael J.
+``Computer Algebra Systems''
+John Wiley and Sons 1999 ISBN 0471983535
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Mignotte 82]{Mig82} Mignotte, Maurice
``Some Useful Bounds''
Computing, Suppl. 4, 259263 (1982), SpringerVerlag
+\bibitem[Wexelblat 87]{Wex87} Wexelblat, Richard L. (ed)
+Proceedings of the SIGPLAN '87 Symposium on
+Interpreter and Interpretive Techniques, St. Paul, Minnesota, June 2426, 1987
+ACM Press, New York, NY 10036, USA, 1987 ISBN 0897912357
+LCCN QA76.7.S54 v22:7 SIGPLAN Notices, vol 22, no 7 (July 1987)
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[McCarthy 83]{McC83} McCarthy G J.
``Investigation into the Multigrid Code MGD1''
Report AERER 10889. Harwell. (1983)
+\bibitem[Wityak 87]{Wit87} Wityak, Sandra
+``Scratchpad II Newsletter''
+Volume 2, Number 1, Nov 1987
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Mie97]{Mie97} Mielenz, Klaus D.
``Computation of Fresnel Integrals''
J. Res. Natl. Inst. Stand. Technol. (NIST) V102 No3 MayJune 1997 pp363365
+\bibitem[WWW1]{WWW1}.
+Software Preservation Group
+\verbwww.softwarepresentation.org/projects/LISP/common_lisp_family
+ keywords = "axiomref",
\end{chunk}
+\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Mie00]{Mie00} Mielenz, Klaus D.
``Computation of Fresnel Integrals II''
J. Res. Natl. Inst. Stand. Technol. (NIST) V105 No4 JulyAug 2000 pp589590
+\bibitem[Yap 00]{Yap00} Yap, Chee Keng
+``Fundamental Problems of Algorithmic Algebra''
+Oxford University Press (2000) ISBN0195125169
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Millen 68]{Mil68} Millen, J. K.
``CHARYBDIS: A LISP program to display mathematical expressions on
typewriterlike devices''
Interactive Systems for Experimental and Applied Mathematics
M. Klerer and J. Reinfelds, eds., Academic Press, New York 1968, pp7990
%\verbaxiomdeveloper.org/axiomwebsite/papers/Mil68.pdf
+\bibitem[Yapp 07]{Yapp07} Yapp, Clifford; Hebisch, Waldek; Kaminski, Kai
+``Literate Programming Tools Implemented in ANSI Common Lisp''
+\verbbrlcad.org/~starseeker/clwebv0.8.lisp.pamphlet
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Minc 79]{Min79} Henryk Minc
``Evaluation of Permanents''
Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 2732.
+\bibitem[Yun 83]{Yun83} Yun, David Y.Y.
+``Computer Algebra and Complex Analysis''
+Computational Aspects of Complex Analysis pp379393
+D. Reidel Publishing Company H. Werner et. al. (eds.)
+ keywords = "axiomref",
\end{chunk}
+\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[More 74]{MGH74} More J J.; Garbow B S.; Hillstrom K E.
``User Guide for Minpack1''
ANL8074 Argonne National Laboratory. (1974)
+\bibitem[Zen92]{Zen92} Zenger, Ch.
+``Gr{\"o}bnerbasen f{\"u}r Differentialformen und ihre
+Implementierung in AXIOM''
+Diplomarbeit, Universit{\"a}t Karlsruhe,
+Karlsruhe, Germany, 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Mikhlin 67]{MS67} Mikhlin S G.; Smolitsky K L.
``Approximate Methods for the Solution of Differential and
Integral Equations''
Elsevier. (1967)
+\bibitem[Zip92]{Zip92} Zippel, Richard
+``Algebraic Computation''
+(unpublished) Cornell University Ithaca, NY Sept 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Mitchell 80]{MG80} Mitchell A R.; Griffiths D F.
``The Finite Difference Method in Partial Differential Equations''
Wiley. (1980)
+\bibitem[Zwi92]{Zwi92} Zwillinger, Daniel
+``Handbook of Integration''
+Jones and Bartlett, 1992, ISBN 0867202939
+ keywords = "axiomref",
\end{chunk}
+\section{Axiom Citations of External Sources}
\begin{chunk}{ignore}
\bibitem[Moler 73]{MS73} Moler C B.; Stewart G W.
``An Algorithm for Generalized Matrix Eigenproblems''
SIAM J. Numer. Anal. 10 241256. 1973
+\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@article{Abla98,
+ author = "Ablamowicz, Rafal",
+ title = "Spinor Representations of Clifford Algebras: A Symbolic Approach",
+ journal = "Computer Physics Communications",
+ volume = "115",
+ number = "23",
+ month = "December",
+ year = "1998",
+ pages = "510535"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Muld97,
 author = "Mulders, Thom",
 title = "A Note on Subresultants and the Lazard/Rioboo/Trager Formula in Rational Function Integration",
 journal = "Journal of Symbolic Computation",
 year = "1997",
 volume = "24",
+@article{Abra06,
+ author = "Abramov, Sergey A.",
+ title = "In Memory of Manuel Bronstein",
+ journal = "Programming and Computer Software",
+ volume = "32",
number = "1",
 month = "July",
 pages = "4550",
 paper = "Muld97.pdf"
+ pages = "5658",
+ publisher = "Pleiades Publishing Inc",
+ year = "2006",
+ paper = "Abra06.pdf"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
An ambiguity in a formula of Lazard, Rioboo and Trager, connecting
subresultants and rational function integration, is indicated and
examples of incorrect interpretations are given.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Munksgaard 80]{Mun80} Munksgaard N.
``Solving Sparse Symmetric Sets of Linear Equations by Preconditioned
Conjugate Gradients''
ACM Trans. Math. Softw. 6 206219. (1980)
+\bibitem[Abramowitz 64]{AS64} Abramowitz, Milton; Stegun, Irene A.
+``Handbook of Mathematical Functions''
+(1964) Dover Publications, NY ISBN 0486612724
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Murray 72]{Mur72} Murray W, (ed)
``Numerical Methods for Unconstrained Optimization''
Academic Press. (1972)
+\bibitem[Abramowitz 68]{AS68} Abramowitz M; Stegun I A
+``Handbook of Mathematical Functions''
+Dover Publications. (1968)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Murtagh 83]{MS83} Murtagh B A.; Saunders M A
``MINOS 5.0 User's Guide''
Report SOL 8320. Department of Operations Research, Stanford University 1983
+\begin{chunk}{axiom.bib}
+@book{Altm05,
+ author = "Altmann, Simon L.",
+ title = "Rotations, Quaternions, and Double Groups",
+ publisher = "Dover Publications, Inc.",
+ year = "2005",
+ isbn = "0486445186"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Musser 78]{Mus78} Musser, David R.
``On the Efficiency of a Polynomial Irreducibility Test''
Journal of the ACM, Vol. 25, No. 2, April 1978, pp. 271282
+\bibitem[Ames 77]{Ames77} Ames W F
+``Nonlinear Partial Differential Equations in Engineering''
+Academic Press (2nd Edition). (1977)
\end{chunk}
\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Nijenhuis 78]{NW78} Nijenhuis and Wilf
``Combinatorical Algorithms''
Academic Press, New York 1978.
+\bibitem[Amos 86]{Amos86} Amos D E
+``Algorithm 644: A Portable Package for Bessel Functions of a Complex
+Argument and Nonnegative Order''
+ACM Trans. Math. Softw. 12 265273. (1986)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Nikolai 79]{Nik79} Nikolai P J.
``Algorithm 538: Eigenvectors and eigenvalues of real generalized
symmetric matrices by simultaneous iteration''
ACM Trans. Math. Softw. 5 118125. (1979)
+\bibitem[Anderson 00]{And00} Anderson, Edward
+``Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem''
+LAPACK Working Note 150, University of Tennessee, UTCS00454,
+December 4, 2000.
\end{chunk}
\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{axiom.bib}
@misc{OCAM14,
 author = "unknown",
 title = "The OCAML website",
 url = "http://ocaml.org"
}
+\begin{chunk}{ignore}
+\bibitem[Anthony 82]{ACH82} Anthony G T; Cox M G; Hayes J G
+``DASL  Data Approximation Subroutine Library''
+National Physical Laboratory. (1982)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ollagnier 94]{Olla94} Ollagnier, Jean Moulin
``Algorithms and methods in differential algebra''
\verbwww.lix.polytechnique.fr/~moulin/papiers/atelier.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Olla94.pdf
+\bibitem[Arnon 88]{Arno88} Arno, D.S.; MIgnotte, M.
+``On Mechanical Quantifier Elimination for Elementary Algebra and Geometry''
+J. Symbolic Computation 5, 237259 (1988)
+\verbhttp://www.sciencedirect.com/science/article/pii/S0747717188800142/
+\verbpdf?md5=62052077d84e6078cc024bc8e29c23c1&
+\verbpid=1s2.0S0747717188800142main.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Arno88.pdf
+ abstract = "
+ We give solutions to two problems of elementary algebra and geometry:
+ (1) find conditions on real numbers $p$, $q$, and $r$ so that the
+ polynomial function $f(x)=x^4+px^2+qx+r$ is nonnegative for all real
+ $x$ and (2) find conditions on real numbers $a$, $b$, and $c$ so that
+ the ellipse $\frac{(xe)^2}{q^2}+\frac{y^2}{b^2}1=0$ lies inside the
+ unit circle $y^2+x^21=0$. Our solutions are obtained by following the
+ basic outline of the method of quantifier elimination by cylindrical
+ algebraic decomposition (Collins, 1975), but we have developed, and
+ have been considerably aided by, modified versions of certain of its
+ steps. We have found three equally simple but not obviously equivalent
+ solutions for the first problem, illustrating the difficulty of
+ obtaining unique ``simplest'' solutions to quantifier elimination
+ problems of elementary algebra and geometry."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Olver 10]{NIST10} Olver, Frank W.; Lozier, Daniel W.;
Boisvert, Ronald F.; Clark, Charles W. (ed)
``NIST Handbook of Mathematical Functions''
(2010) Cambridge University Press ISBN 9780521192255
+\begin{chunk}{axiom.bib}
+@article{Aubr99,
+ author = "Aubry, Phillippe and Lazard, Daniel and {Moreno Maza}, Marc",
+ title = "On the Theories of Triangular Sets",
+ year = "1999",
+ pages = "105124",
+ journal = "Journal of Symbolic Computation",
+ volume = "28",
+ url = "http://www.csd.uwo.ca/~moreno/Publications/AubryLazardMorenoMaza1999JSC.pdf",
+ papers = "Aubr99.pdf",
+ abstract = "
+ Different notions of triangular sets are presented. The relationship
+ between these notions are studied. The main result is that four
+ different existing notions of {\sl good} triangular sets are
+ equivalent."
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[OpenM]{OpenM}.
``OpenMath Technical Overview''
\verbwww.openmath.org/overview/technical.html
+\bibitem[Aubry 96]{Aub96} Aubry, Philippe; Maza, Marc Moreno
+``Triangular Sets for Solving Polynomial Systems: a Comparison of Four Methods''
+\verbwww.lip6.fr/lip6/reports/1997/lip6.1997.009.ps.gz
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Aub96.ps
+ abstract = "
+ Four methods for solving polynomial systems by means of triangular
+ sets are presented and implemented in a unified way. These methods are
+ those of Wu, Lazard, Kalkbrener, and Wang. They are compared on
+ various examples with emphasis on efficiency, conciseness and
+ legibility of the outputs."
\end{chunk}
+\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Ortega 70]{OR70} Ortega J M.; Rheinboldt W C.
``Iterative Solution of Nonlinear Equations in Several Variables''
Academic Press. (1970)
+\bibitem[Bailey 66]{Bai66} Bailey P B
+``SturmLiouville Eigenvalues via a Phase Function''
+SIAM J. Appl. Math . 14 242249. (1966)
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Ostr1845,
 author = "Ostrogradsky. M.W.",
 title = "De l'int\'{e}gration des fractions rationelles.",
 journal = "Bulletin de la Classe PhysicoMath\'{e}matiques de l'Acae\'{e}mie Imp\'{e}riale des Sciences de St. P\'{e}tersbourg,",
 volume = "IV",
 pages = "145167,286300",
 year = "1845"
}
+\begin{chunk}{ignore}
+\bibitem[Baker 96]{BGM96} Baker, George A.; GravesMorris, Peter
+``Pade Approximants''
+Cambridge University Press, March 1996 ISBN 9870521450072
\end{chunk}
\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Paige 75]{PS75} Paige C C.; Saunders M A.
``Solution of Sparse Indefinite Systems of Linear Equations''
SIAM J. Numer. Anal. 12 617629. (1975)
+\begin{chunk}{ignore}
+\bibitem[Baker 10]{Ba10} Baker, Martin
+``3D World Simulation''
+\verbwww.euclideanspace.com
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Paige 82a]{PS82a} Paige C C.; Saunders M A.
``LSQR: An Algorithm for Sparse Linear Equations and Sparse Leastsquares''
ACM Trans. Math. Softw. 8 4371. (1982)
+\begin{chunk}{axiom.bib}
+@misc{Bake14,
+ author = "Baker, Martin",
+ title = "Axiom Architecture",
+ year = "2014",
+ url = "http://www.euclideanspace.com/prog/scratchpad/internals/ccode"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Paige 82b]{PS82b} Paige C C.; Saunders M A.
``ALGORITHM 583 LSQR: Sparse Linear Equations and Leastsquares Problems''
ACM Trans. Math. Softw. 8 195209. (1982)
+\bibitem[Banks 68]{BK68} Banks D O; Kurowski I
+``Computation of Eigenvalues of Singular SturmLiouville Systems''
+Math. Computing. 22 304310. (1968)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Parker 84]{Par84} Parker, R. A.
``The Computer Calculation of Modular Characters (The MeatAxe)''
M. D. Atkinson (Ed.), Computational Group Theory
Academic Press, Inc., London 1984
+\bibitem[Bard 74]{Bard74} Bard Y
+``Nonlinear Parameter Estimation''
+Academic Press. 1974
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Parlett 80]{Par80} Parlett B N.
``The Symmetric Eigenvalue Problem''
PrenticeHall. 1980
+\bibitem[Barrodale 73]{BR73} Barrodale I; Roberts F D K
+``An Improved Algorithm for Discrete $ll_1$ Linear Approximation''
+SIAM J. Numer. Anal. 10 839848. (1973)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Parnas 10]{PJ10} Parnas, David Lorge; Jin, Ying
``Defining the meaning of tabular mathematical expressions''
Science of Computer Programming V75 No.11 Nov 2010 pp9801000 Elesevier
+\bibitem[Barrodale 74]{BR74} Barrodale I; Roberts F D K
+``Solution of an Overdetermined System of Equations in the $ll_1norm$.''
+Comm. ACM. 17, 6 319320. (1974)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Parnas 95]{PM95} Parnas, David Lorge; Madey, Jan
``Functional Documents for Computer Systems''
Science of Computer Programming V25 No.1 Oct 1995 pp4161 Elesevier
+\bibitem[Beauzamy 92]{Bea92} Beauzamy, Bernard
+``Products of polynomials and a priori estimates for
+coefficients in polynomial decompositions: a sharp result''
+J. Symbolic Computation (1992) 13, 463472
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bea92.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Paul 81]{Paul81} Paul, Richard
``Robot Manipulators''
MIT Press 1981
+\bibitem[Beauzamy 93]{Bea93} Beauzamy, Bernard; Trevisan, Vilmar;
+Wang, Paul S.
+``Polynomial Factorization: Sharp Bounds, Efficient Algorithms''
+J. Symbolic Computation (1993) 15, 393413
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bea93.pdf
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Pear56,
 author = "Pearcey, T.",
 title = "Table of the Fresnel Integral",
 publisher = "Cambridge University Press",
 year = "1956"
+@article{Bert95,
+ author = "Bertrand, Laurent",
+ title = "Computing a hyperelliptic integral using arithmetic in the
+ jacobian of the curve",
+ journal = "Applicable Algebra in Engineering, Communication and Computing",
+ volume = "6",
+ pages = "275298",
+ year = "1995",
+ abstract = "
+ In this paper, we describe an efficient algorithm for computing an
+ elementary antiderivative of an algebraic function defined on a
+ hyperelliptic curve. Our algorithm combines B.M. Trager's integration
+ algorithm and a technique for computing in the Jacobian of a
+ hyperelliptic curve introduced by D.G. Cantor. Our method has been
+ implemented and successfully compared to Trager's general algorithm."
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Pereyra 79]{Per79} Pereyra V.
``PASVA3: An Adaptive FiniteDifference Fortran Program for First Order
Nonlinear, Ordinary Boundary Problems''
Codes for Boundary Value Problems in Ordinary Differential Equations.
Lecture Notes in Computer Science.
(ed B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76
SpringerVerlag. (1979)

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Peters 67a]{Pet67a} Peters G.
``NPL Algorithms Library''
Document No. F2/03/A. (1967)
+\bibitem[Berzins 87]{BBG87} Berzins M; Brankin R W; Gladwell I.
+``Design of the Stiff Integrators in the NAG Library''
+Technical Report. TR14/87 NAG. (1987)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Peters 67b]{Pet67b} Peters G.
``NPL Algorithms Library''
Document No.F1/04/A (1967)
+\bibitem[Berzins 90]{Ber90} Berzins M
+``Developments in the NAG Library Software for Parabolic Equations''
+Scientific Software Systems. (ed J C Mason and M G Cox)
+Chapman and Hall. 5972. (1990)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Peters 70]{PW70} Peters G.; Wilkinson J H.
``The Leastsquares Problem and Pseudoinverses''
Comput. J. 13 309316. (1970)
+\bibitem[Birkhoff 62]{BR62} Birkhoff, G; Rota, G C
+``Ordinary Differential Equations''
+Ginn \& Co., Boston and New York. (1962)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Peters 71]{PW71} Peters G.; Wilkinson J H.
``Practical Problems Arising in the Solution of Polynomial Equations''
J. Inst. Maths Applics. 8 1635. (1971)
+\bibitem[Boyd9 3a]{Boyd93a} Boyd, David W.
+``Bounds for the Height of a Factor of a Polynomial in
+Terms of Bombieri's Norms: I. The Largest Factor''
+J. Symbolic Computation (1993) 16, 115130
+%\verbaxiomdeveloper.org/axiomwebsite/Boyd93a.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Pierce 82]{Pie82} R.S. Pierce
``Associative Algebras''
Graduate Texts in Mathematics 88
SpringerVerlag, Heidelberg, 1982, ISBN 0387906932
+\bibitem[Boyd 93b]{Boyd93b} Boyd, David W.
+``Bounds for the Height of a Factor of a Polynomial in
+Terms of Bombieri's Norms: II. The Smallest Factor''
+J. Symbolic Computation (1993) 16, 131145
+%\verbaxiomdeveloper.org/axiomwebsite/Boyd93b.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Piessens 73]{Pie73} Piessens R.
``An Algorithm for Automatic Integration''
Angewandte Informatik. 15 399401. (1973)
+\bibitem[Braman 02a]{BBM02a} Braman, K.; Byers, R.; Mathias, R.
+``The MultiShift QR Algorithm Part I: Maintaining Well Focused Shifts,
+and Level 3 Performance''
+SIAM Journal of Matrix Analysis, volume 23, pages 929947, 2002.
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Piessens 74]{PMB74} Piessens R.;; Mertens I.; Branders M.
``Integration of Functions having Endpoint Singularities''
Angewandte Informatik. 16 6568. (1974)
+\bibitem[Braman 02b]{BBM02b} Braman, K.; Byers, R.; Mathias, R.
+``The MultiShift QR Algorithm Part II: Aggressive Early Deflation''
+SIAM Journal of Matrix Analysis, volume 23, pages 948973, 2002.
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Piessens 75]{PB75} Piessens R.; Branders M.
``Algorithm 002. Computation of Oscillating Integrals''
J. Comput. Appl. Math. 1 153164. (1975)
+\bibitem[Brent 75]{Bre75} Brent, R. P.
+``MultiplePrecision ZeroFinding Methods and the Complexity
+of Elementary Function Evaluation, Analytic Computational Complexity''
+J. F. Traub, Ed., Academic Press, New York 1975, 151176
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Piessens 76]{PVRBM76} Piessens R.; Van RoyBranders M.; Mertens I.
``The Automatic Evaluation of Cauchy Principal Value Integrals''
Angewandte Informatik. 18 3135. (1976)
+\bibitem[Brent 78]{BK78} Brent, R. P.; Kung, H. T.
+``Fast Algorithms for Manipulating Formal Power Series''
+Journal of the Association for Computing Machinery,
+Vol. 25, No. 4, October 1978, 581595
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Piessens 83]{PDUK83} Piessens R.; De DonckerKapenga E.;
Uberhuber C.; Kahaner D.
``QUADPACK, A Subroutine Package for Automatic Integration''
SpringerVerlag.(1983)
+\bibitem[Brigham 73]{Bri73} Brigham E O
+``The Fast Fourier Transform''
+PrenticeHall. (1973)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Polya 37]{Pol37} Polya, G.
``Kombinatorische Anzahlbestimmungen fur Gruppen,
Graphen und chemische Verbindungen''
Acta Math. 68 (1937) 145254.
+\bibitem[Brillhart 69]{Bri69} Brillhart, John
+``On the Euler and Bernoulli polynomials''
+J. Reine Angew. Math., v. 234, (1969), pp. 4564
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Powell 70]{Pow70} Powell M J D.
``A Hybrid Method for Nonlinear Algebraic Equations''
Numerical Methods for Nonlinear Algebraic Equations.
(ed P Rabinowitz) Gordon and Breach. (1970)
+\bibitem[Brillhart 90]{Bri90} Brillhart, John
+``Note on Irreducibility Testing''
+Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 13791381
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Powell 74]{Pow74} Powell M J D.
``Introduction to Constrained Optimization''
Numerical Methods for Constrained Optimization.
(ed P E Gill and W Murray) Academic Press. pp128. 1974
+\bibitem[Bronstein 98a]{Bro98a} Bronstein, M.; Grabmeier, J.; Weispfenning, V. (eds)
+``Symbolic Rewriting Techniques''
+Progress in Computer Science and Applied Logic 15, BirkhauserVerlag, Basel
+ISBN 3764359013 (1998)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Powell 83]{Pow83} Powell M J D.
``Variable Metric Methods in Constrained Optimization''
Mathematical Programming: The State of the Art.
(ed A Bachem, M Groetschel and B Korte) SpringerVerlag. pp288311. 1983
+\bibitem[Bronstein 88]{Bro88} Bronstein, Manual
+``The Transcendental Risch Differential Equation''
+J. Symbolic Computation (1990) 9, pp4960 Feb 1988
+IBM Research Report RC13460 IBM Corp. Yorktown Heights, NY
+\verbwww.sciencedirect.com/science/article/pii/S0747717108800065
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro88.pdf
+ abstract = "
+ We present a new rational algorithm for solving Risch differential
+ equations in towers of transcendental elementary extensions. In
+ contrast to a recent algorithm by Davenport we do not require a
+ progressive reduction of the denominators involved, but use weak
+ normality to obtain a formula for the denominator of a possible
+ solution. Implementation timings show this approach to be faster than
+ a Hermitelike reduction."
\end{chunk}
\begin{chunk}{axiom.bib}
@inproceedings{Prat73,
 author = "Pratt, Vaughan R.",
 title = "Top down operator precedence",
 booktitle = "Proc. 1st annual ACM SIGACTSIGPLAN Symposium on Principles of Programming Languages",
 series = "POPL'73",
 pages = "4151",
 year = "1973",
 url = "http://hall.org.ua/halls/wizzard/pdf/Vaughan.Pratt.TDOP.pdf",
 keywords = "axiomref",
 paper = "Prat73.pdf"
+@techreport{Bron98,
+ author = "Bronstein, Manuel",
+ title = "The lazy hermite reduction",
+ type = "Rapport de Recherche",
+ number = "RR3562",
+ year = "1998",
+ institution = "French Institute for Research in Computer Science",
+ paper = "Bron98.pdf",
+ abstract = "
+ The Hermite reduction is a symbolic integration technique that reduces
+ algebraic functions to integrands having only simple affine
+ poles. While it is very effective in the case of simple radical
+ extensions, its use in more general algebraic extensions requires the
+ precomputation of an integral basis, which makes the reduction
+ impractical for either multiple algebraic extensions or complicated
+ ground fields. In this paper, we show that the Hermite reduction can
+ be performed without {\sl a priori} computation of either a primitive
+ element or integral basis, computing the smallest order necessary for
+ a particular integrand along the way."
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Press 95]{PTVF95} Press, William H.; Teukolsky, Saul A.;
Vetterling, William T.; Flannery, Brian P.
``Numerical Recipes in C''
Cambridge University Press (1995) ISBN 0521431085

\end{chunk}

\begin{chunk}{ignore}
\bibitem[Pryce 77]{PH77} Pryce J D.; Hargrave B A.
``The Scale Pruefer Method for oneparameter and multiparameter eigenvalue
problems in ODEs''
Inst. Math. Appl., Numerical Analysis Newsletter. 1(3) (1977)
+\begin{chunk}{axiom.bib}
+@misc{Bro98b,
+ author = "Bronstein, Manuel",
+ title = "Symbolic Integration Tutorial",
+ series = "ISSAC'98",
+ year = "1998",
+ address = "INRIA Sophia Antipolis",
+ url =
+ "http://wwwsop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf",
+ paper = "Bro98b.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Pryce 81]{Pry81} Pryce J D.
``Two codes for SturmLiouville problems''
Technical Report CS8101. Dept of Computer Science, Bristol University (1981)
+\bibitem[Brown 99]{Brow99} Brown, Christopher W.
+``Solution Formula Construction for Truth Invariant CADs''
+Ph.D Thesis, Univ. Delaware (1999)
+\verbwww.usna.edu/Users/cs/wcbrown/research/thesis.ps.gz
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Brow99.pdf
+ abstract = "
+ The CADbased quantifier elimination algorithm takes a formula from
+ the elementary theory of real closed fields as input, and constructs a
+ CAD of the space of the formula's unquantified variables. This
+ decomposition is truth invariant with respect to the input formula,
+ meaning that the formula is either identically true or identically
+ false in each cell of the decomposition. The method determines the
+ truth of the input formula for each cell of the CAD, and then uses the
+ CAD to construct a solution formula  a quantifier free formula that
+ is equivalent to the input formula. This final phase of the algorithm,
+ the solution formula construction phase, is the focus of this thesis.
+
+ An optimal solution formula construction algorithm would be {\sl
+ complete}  i.e. applicable to any truthinvariant CAD, would be {\sl
+ efficient}, and would produce {\sl simple} solution formulas. Prior to
+ this thesis, no method was available with even two of these three
+ properties. Several algorithms are presented, all addressing problems
+ related to solution formula construction. In combination, these
+ provide an efficient and complete method for constructing solution
+ formulas that are simple in a variety of ways.
+
+ Algorithms presented in this thesis have been implemented using the
+ SACLIB library, and integrated into QEPCAD, a SACLIBbased
+ implementation of quantifier elimination by CAD. Example computations
+ based on these implementations are discussed."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Pryce 86]{Pry86} Pryce J D.
``Error Estimation for Phasefunction Shooting Methods for
SturmLiouville Problems''
J. Num. Anal. 6 103123. (1986)
+\bibitem[Brown 02]{Brow02} Brown, Christopher W.
+``QEPCAD B  A program for computing with semialgebraic sets using CADs''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Brow02.pdf
+ abstract = "
+ This report introduces QEPCAD B, a program for computing with real
+ algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD
+ B both extends and improves upon the QEPCAD system for quantifier
+ elimination by partial cylindrical algebraic decomposition written by
+ Hoon Hong in the early 1990s. This paper briefly discusses some of the
+ improvements in the implementation of CAD and quantifier elimination
+ vis CAD, and provides somewhat more detail on extensions to the system
+ that go beyond quantifier elimination. The author is responsible for
+ most of the extended features of QEPCAD B, but improvements to the
+ basic CAD implementation and to the SACLIB library on which QEPCAD is
+ based are the results of many people's work."
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Puff09,
 author = "Puffinware LLC",
 title = "Singular Value Decomposition (SVD) Tutorial",
 url = "http://www.puffinwarellc.com/p3a.htm"
}

\end{chunk}

\subsection{Q} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[QuintanaOrti 06]{QG06} QuintanaOrti, Gregorio;
van de Geijn, Robert
``Improving the performance of reduction to Hessenberg form''
ACM Transactions on Mathematical Software, 32(2):180194, June 2006.
+@article{Burg74,
+ author = "William H. Burge",
+ title = "Stream Processing Functions",
+ year = "1974",
+ month = "January",
+ journal = "IBM Journal of Research and Development",
+ volume = "19",
+ issue = "1",
+ pages = "1225",
+ papers = "Burg74.pdf",
+ abstract = "
+ One principle of structured programming is that a program should be
+ separated into meaningful independent subprograms, which are then
+ combined so that the relation of the parts to the whole can be clearly
+ established. This paper describes several alternative ways to compose
+ programs. The main method used is to permit the programmer to denote
+ by an expression the sequence of values taken on by a variable. The
+ sequence is represented by a function called a stream, which is a
+ functional analog of a coroutine. The conventional while and for loops
+ of structured programming may be composed by a technique of stream
+ processing (analogous to list processing), which results in more
+ structured programs than the orignals. This technique makes it
+ possible to structure a program in a natural way into its logically
+ separate parts, which can then be considered independently."
+}
\end{chunk}
\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Rabinowitz 70]{Rab70} Rabinowitz P.
``Numerical Methods for Nonlinear Algebraic Equations''
Gordon and Breach. (1970)
+\bibitem[Carlson 65]{Car65} Carlson B C
+``On Computing Elliptic Integrals and Functions''
+J Math Phys. 44 3651. (1965)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ralston 65]{Ral65} Ralston A.
``A First Course in Numerical Analysis''
McGrawHill. 8790. (1965)
+\bibitem[Carlson 77a]{Car77a} Carlson B C
+``Elliptic Integrals of the First Kind''
+SIAM J Math Anal. 8 231242. (1977)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ramakrishnan 03]{Ram03} Ramakrishnan, Maya
``A Gentle Introduction to Lyapunov Functions''
ORSUM August 2003
\verbwww.or.ms.unimelb.edu.au/handouts/lyaptalk.1.pdf
+\bibitem[Carlson 77b]{Car77b} Carlson B C
+``Special Functions of Applied Mathematics''
+Academic Press. (1977)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ramsey 03]{Ra03} Ramsey, Norman
``NowebA Simple, Extensible Tool for Literate Programming''
\verbwww.eecs.harvard.edu/~nr/noweb
+\bibitem[Carlson 78]{Car78} Carlson B C,
+``Computing Elliptic Integrals by Duplication''
+(Preprint) Department of Physics, Iowa State University. (1978)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Redfield 27]{Red27} Redfield, J.H.
``The Theory of GroupReduced Distributions''
American J. Math., 49 (1927) 433455.
+\bibitem[Carlson 88]{Car88} Carlson B C,
+``A Table of Elliptic Integrals of the Third Kind''
+Math. Comput. 51 267280. (1988)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Reinsch 67]{Rei67} Reinsch C H.
``Smoothing by Spline Functions''
Num. Math. 10 177183. (1967)
+\bibitem[Cauchy 1829]{Cau1829} AugustinLux Cauchy
+``Exercices de Math\'ematiques Quatri\`eme Ann\'ee. De Bure Fr\`eres''
+Paris 1829 (reprinted Oeuvres, II S\'erie, Tome IX,
+GauthierVillars, Paris, 1891).
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Renka 84]{Ren84} Renka R L.
``Algorithm 624: Triangulation and Interpolation of Arbitrarily Distributed
Points in the Plane''
ACM Trans. Math. Softw. 10 440442. (1984)
+\bibitem[Ch\`eze 07]{Chez07} Ch\'eze, Guillaume; Lecerf, Gr\'egoire
+``Lifting and recombination techniques for absolute factorization''
+Journal of Complexity, VOl 23 Issue 3 June 2007 pp 380420
+\verbwww.sciencedirect.com/science/article/pii/S0885064X07000465
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Chez07.pdf
+ abstract = "
+ In the vein of recent algorithmic advances in polynomial factorization
+ based on lifting and recombination techniques, we present new faster
+ algorithms for computing the absolute factorization of a bivariate
+ polynomial. The running time of our probabilistic algorithm is less
+ than quadratic in the dense size of the polynomial to be factored."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Renka 84]{RC84} Renka R L.; Cline A K.
``A Trianglebased C Interpolation Method''
Rocky Mountain J. Math. 14 223237. (1984)
+\bibitem[Childs 79]{CSDDN79} Childs B; Scott M; Daniel J W; Denman E;
+Nelson P (eds)
+``Codes for Boundaryvalue Problems in Ordinary Differential Equations''
+Lecture Notes in Computer Science. 76 (1979) SpringerVerlag
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Reutenauer 93]{Re93} Reutenauer, Christophe
``Free Lie Algebras''
Oxford University Press, June 1993 ISBN 0198536798
+\bibitem[Clausen 89]{Cla89} Clausen, M.; Fortenbacher, A.
+``Efficient Solution of Linear Diophantine Equations''
+JSC (1989) 8, 201216
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Reznick 93]{Rezn93} Reznick, Bruce
``An Inequality for Products of Polynomials''
Proc. AMS Vol 117 No 4 April 1993
%\verbaxiomdeveloper.org/axiomwebsite/papers/Rezn93.pdf
+\bibitem[Clenshaw 55]{Cle55} Clenshaw C W,
+``A Note on the Summation of Chebyshev Series''
+Math. Tables Aids Comput. 9 118120. (1955)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rich xx]{Rixx} Rich, A.D.; Jeffrey, D.J.
``Crafting a Repository of Knowledge Based on Transformation''
\verbwww.apmaths.uwo.ca/~djeffrey/Offprints/IntegrationRules.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Rixx.pdf
+\bibitem[Clenshaw 60]{Cle60} Clenshaw C W
+``Curve Fitting with a Digital Computer''
+Comput. J. 2 170173. (1960)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe the development of a repository of mathematical knowledge
based on transformation rules. The specific mathematical problem is
indefinite integration. It is important that the repository be not
confused with a lookup table. The database of transformation rules is
at present encoded in Mathematica, but this is only one convenient
form of the repository, and it could be readily translated into other
formats. The principles upon which the set of rules is compiled is
described. One important principle is minimality. The benefits of the
approach are illustrated with examples, and with the results of
comparisons with other approaches.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Rich 10]{Ri10} Rich, Albert D.
``Rulebased Mathematics''
\verbwww.apmaths.uwo.ca/~arich
+\bibitem[Clenshaw 62]{Cle62} Clenshaw C W
+``Mathematical Tables. Chebyshev Series for Mathematical Functions''
+HMSO. (1962)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Richardson 94]{RF94} Richardson, Dan; Fitch, John
``The identity problem for elementary functions and constants''
ACM Proc. of ISSAC 94 pp285290 ISBN 0897916387
+\bibitem[Cline 84]{CR84} Cline A K; Renka R L,
+``A Storageefficient Method for Construction of a Thiessen Triangulation''
+Rocky Mountain J. Math. 14 119139. (1984)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Richtmyer 67]{RM67} Richtmyer R D.; Morton K W.
``Difference Methods for Initialvalue Problems''
Interscience (2nd Edition). (1967)
+\bibitem[Conway 87]{CCNPW87} Conway, J.; Curtis, R.; Norton, S.; Parker, R.;
+Wilson, R.
+``Atlas of Finite Groups''
+Oxford, Clarendon Press, 1987
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rioboo 92]{REFRio92} Rioboo, R.
``Real algebraic closure of an ordered field, implementation in Axiom''
In Wang [Wan92], pp206215 ISBN 0897914899 (soft cover)
In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206215.
0897914902 (hard cover) LCCN QA76.95.I59 1992
+\bibitem[Conway 03]{CS03} Conway, John H.; Smith, Derek, A.
+``On Quaternions and Octonions''
+A.K Peters, Natick, MA. (2003) ISBN 1568811349
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rioboo 96]{Rio96} Rioboo, R.
``Generic computation of the real closure of an ordered field''
In Mathematics and Computers in Simulation Volume 42, Issue 46,
November 1996.
+\bibitem[Cox 72]{Cox72} Cox M G
+``The Numerical Evaluation of Bsplines''
+J. Inst. Math. Appl. 10 134149. (1972)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ritt 50]{Ritt50} Ritt, Joseph Fels
``Differential Algebra''
AMS Colloquium Publications Volume 33 ISBN 9780821846384
+\bibitem[CH 73]{CH73} Cox M G; Hayes J G
+``Curve fitting: a guide and suite of algorithms for the
+nonspecialist user''
+Report NAC26. National Physical Laboratory. (1973)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rote 01]{Rote01} Rote, G\"unter
``Divisionfree algorithms for the determinant and the Pfaffian''
in Computational Discrete Mathematics ISBN 3540427759 pp119135
\verbpage.mi.fuberlin.de/rote/Papers/pdf/Divisionfree+algorithms.pdf
+\bibitem[Cox 74a]{Cox74a} Cox M G
+``A Datafitting Package for the Nonspecialist User''
+Software for Numerical Mathematics. (ed D J Evans) Academic Press. (1974)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rubey 07]{Rub07} Rubey, Martin
``Formula Guessing with Axiom''
April 2007
+\bibitem[Cox 74b]{Cox74b} Cox M G
+``Numerical methods for the interpolation and approximation of data
+by spline functions''
+PhD Thesis. City University, London. (1975)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rutishauser 69]{Rut69} Rutishauser H.
``Computational aspects of F L Bauer's simultaneous iteration method''
Num. Math. 13 413. (1969)
+\bibitem[Cox 75]{Cox75} Cox M G
+``An Algorithm for Spline Interpolation''
+J. Inst. Math. Appl. 15 95108. (1975)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rutishauser 70]{Rut70} Rutishauser H.
``Simultaneous iteration method for symmetric matrices''
Num. Math. 16 205223. (1970)
+\bibitem[Cox 77]{Cox77} Cox M G
+``A Survey of Numerical Methods for Data and Function Approximation''
+The State of the Art in Numerical Analysis. (ed D A H Jacobs)
+Academic Press. 627668. (1977)
+ keywords = "survey",
\end{chunk}
\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Schafer 66]{Sch66} Schafer, R.D.
``An Introduction to Nonassociative Algebras''
Academic Press, New York, 1966
+\bibitem[Cox 78]{Cox78} Cox M G
+``The Numerical Evaluation of a Spline from its Bspline Representation''
+J. Inst. Math. Appl. 21 135143. (1978)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Schoenberg 53]{SW53} Schoenberg I J.; Whitney A.
``On Polya Frequency Functions III''
Trans. Amer. Math. Soc. 74 246259. (1953)
+\bibitem[Curtis 74]{CPR74} Curtis A R; Powell M J D; Reid J K
+``On the Estimation of Sparse Jacobian Matrices''
+J. Inst. Maths Applics. 13 117119. (1974)
\end{chunk}
+\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Schoenhage 82]{Sch82} Schoenhage, A.
``The fundamental theorem of algebra in terms of computational complexity''
preliminary report, Univ. Tuebingen, 1982
+\bibitem[Dahlquist 74]{DB74} Dahlquist G; Bjork A
+``Numerical Methods''
+Prentice Hall. (1974)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Schonfelder 76]{Sch76} Schonfelder J L.
``The Production of Special Function Routines for a MultiMachine Library''
Software Practice and Experience. 6(1) (1976)
+\bibitem[Dalmas 98]{DA98} Dalmas, Stephane; Arsac, Olivier
+``The INRIA OpenMath Library''
+Projet SAFIR, INRIA Sophia Antipolis Nov 25, 1998
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Segg93,
 author = "{von Seggern}, David Henry",
 title = "CRC Standard Curves and Surfaces",
 publisher = "CRC Press",
 year = "1993",
 isbn = "0849301963"
}
+\begin{chunk}{ignore}
+\bibitem[Dantzig 63]{Dan63} Dantzig G B
+``Linear Programming and Extensions''
+Princeton University Press. (1963)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Seiler 95a]{Sei95a} Seiler, W.M.; Calmet, J.
``JET  An Axiom Environment for Geometric Computations with Differential
Equations''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Sei95a.pdf
+\bibitem[Davenport]{Dav} Davenport, James
+``On Brillhart Irreducibility.''
+To appear.
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Shepard 68]{She68} Shepard D.
``A Twodimensional Interpolation Function for Irregularly Spaced Data''
Proc. 23rd Nat. Conf. ACM. Brandon/Systems Press Inc.,
Princeton. 517523. 1968
+\bibitem[Davenport 93]{RefDav93} Davenport, J.H.
+``Primality testing revisited''
+Technical Report TR2/93
+(ATR/6)(NP2556) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA
+and Oxford, UK, August 1993
+\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Shirayanagi 96]{Shir96} Shirayanagi, Kiyoshi
``Floating point Gr\"obner bases''
Mathematics and Computers in Simulation 42 pp 509528 (1996)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Shir96.pdf
+\bibitem[Davis 67]{DR67} Davis P J; Rabinowitz P
+``Numerical Integration''
+Blaisdell Publishing Company. 3352. (1967)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Bracket coefficients for polynomials are introduced. These are like
specific precision floating point numbers together with error
terms. Working in terms of bracket coefficients, an algorithm that
computes a Gr\"obner basis with floating point coefficients is
presented, and a new criterion for determining whether a bracket
coefficient is zero is proposed. Given a finite set $F$ of polynomials
with real coefficients, let $G_\mu$ be the result of the algorithm for
$F$ and a precision $\mu$, and $G$ be a true Gr\"obner basis of
$F$. Then, as $\mu$ approaches infinity, $G_\mu$ converges to $G$
coefficientwise. Moreover, there is a precision $M$ such that if
$\mu \ge M$, then the sets of monomials with nonzero coefficients of
$G_\mu$ and $G$ are exactly the same. The practical usefulness of the
algorithm is suggested by experimental results.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Sims 71]{Sims71} Sims, C.
``Determining the Conjugacy Classes of a Permutation Group''
Computers in Algebra and Number Theory, SIAMAMS Proc., Vol. 4,
American Math. Soc., 1991, pp191195
+\bibitem[Davis 75]{DR75} Davis P J; Rabinowitz P
+``Methods of Numerical Integration''
+Academic Press. (1975)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Singer 89]{Sing89} Singer, M.F.
``Formal Solutions of Differential Equations''
J. Symbolic COmputation 10, No.1 5994 (1990)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Sing89.pdf
 keywords = "survey",
+\bibitem[DeBoor 72]{DeB72} De Boor C
+``On Calculating with Bsplines''
+J. Approx. Theory. 6 5062. (1972)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We give a survey of some methods for finding formal solutions of
differential equations. These include methods for finding power series
solutions, elementary and liouvillian solutions, first integrals, Lie
theoretic methods, transform methods, asymptotic methods. A brief
discussion of difference equations is also included.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Sit 92]{REFSit92} Sit, William
``An Algorithm for Parametric Linear Systems''
J. Sym. Comp., April 1992
+\bibitem[De Doncker 78]{DeD78} De Doncker E,
+``An Adaptive Extrapolation Algorithm for Automatic Integration''
+Signum Newsletter. 13 (2) 1218. (1978)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Smith 67]{Smi67} Smith B T.
``ZERPOL: A Zero Finding Algorithm for Polynomials Using Laguerre's Method''
Technical Report. Department of Computer Science, University of Toronto,
Canada. (1967)
+\bibitem[Demmel 89]{Dem89} Demmel J W
+``On Floatingpoint Errors in Cholesky''
+LAPACK Working Note No. 14. University of Tennessee, Knoxville. 1989
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Smith 85]{Smi85} Smith G D.
``Numerical Solution of Partial Differential Equations: Finite Difference
Methods''
Oxford University Press (3rd Edition). (1985)
+\bibitem[Dennis 77]{DM77} Dennis J E Jr; More J J
+``QuasiNewton Methods, Motivation and Theory''
+SIAM Review. 19 4689. 1977
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Sobol 74]{Sob74} Sobol I M.
``The Monte Carlo Method''
The University of Chicago Press. 1974
+\bibitem[Dennis 81]{DS81} Dennis J E Jr; Schnabel R B
+``A New Derivation of Symmetric PositiveDefinite Secant Updates''
+Nonlinear Programming 4. (ed O L Mangasarian, R R Meyer and S M. Robinson)
+Academic Press. 167199. (1981)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Steele 90]{Ste90} Steele, Guy L.
``Common Lisp The Language''
Second Edition ISBN 1555580416 Digital Press (1990)
+\bibitem[Dennis 83]{DS83} Dennis J E Jr; Schnabel R B
+``Numerical Methods for Unconstrained Optimixation and Nonlinear Equations''
+PrenticeHall.(1983)
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Stic93,
 author = "Stichtenoth, H.",
 title = "Algebraic function fields and codes",
 publisher = "SpringerVerlag",
 year = "1993"
}
+\begin{chunk}{ignore}
+\bibitem[Dierckx 75]{Die75} Dierckx P
+``An Algorithm for Smoothing, Differentiating and Integration of
+Experimental Data Using Spline Functions''
+J. Comput. Appl. Math. 1 165184. (1975)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Stinson 90]{Stin90} Stinson, D.R.
``Some observations on parallel Algorithms for fast exponentiation
in $GF(2^n)$''
Siam J. Comp., Vol.19, No.4, pp.711717, August 1990
%\verbaxiomdeveloper.org/axiomwebsite/Stin90.pdf
+\bibitem[Dierckx 81]{Die81} Dierckx P
+``An Improved Algorithm for Curve Fitting with Spline Functions''
+Report TW54. Dept. of Computer Science, Katholieke Universiteit Leuven. 1981
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A normal basis represention in $GF(2^n)$ allows squaring to be
accomplished by a cyclic shift. Algorithms for multiplication in
$GF(2^n)$ using a normal basis have been studied by several
researchers. In this paper, algorithms for performing exponentiation
in $GF(2^n)$ using a normal basis, and how they can be speeded up by
using parallelization, are investigated.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Stroud 66]{SS66} Stroud A H.; Secrest D.
``Gaussian Quadrature Formulas''
PrenticeHall. (1966)
+\bibitem[Dierckx 82]{Die82} Dierckx P
+``A Fast Algorithm for Smoothing Data on a Rectangular Grid while using
+Spline Functions''
+SIAM J. Numer. Anal. 19 12861304. (1982)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Stroud 71]{Str71} Stroud A H.
``Approximate Calculation of Multiple Integrals''
PrenticeHall 1971
+\bibitem[Dongarra 79]{DMBS79} Dongarra J J; Moler C B; Bunch J R;
+Stewart G W
+``LINPACK Users' Guide''
+SIAM, Philadelphia. (1979)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Swarztrauber 79]{SS79} Swarztrauber P N.; Sweet R A.
``Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial
Differential Equations''
ACM Trans. Math. Softw. 5 352364. (1979)
+\bibitem[Dongarra 85]{DCHH85} Dongarra J J; Du Croz J J; Hammarling S;
+Hanson R J
+``A Proposal for an Extended set of Fortran Basic Linear
+Algebra Subprograms''
+SIGNUM Newsletter. 20 (1) 218. (1985)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Swarztrauber 84]{SS84} Swarztrauber P N.
``Fast Poisson Solvers''
Studies in Numerical Analysis. (ed G H Golub)
Mathematical Association of America. (1984)
+\bibitem[Dongarra 88]{REFDON88} Dongarra, Jack J.; Du Croz, Jeremy;
+Hammarling, Sven; Hanson, Richard J.
+``An Extended Set of FORTRAN Basic Linear Algebra Subroutines''
+ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988,
+pp 117
\end{chunk}
\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{axiom.bib}
@book{Tait1890,
 author = "Tait, P.G.",
 title = "An Elementary Treatise on Quaternions",
 publisher = "C.J. Clay and Sons, Cambridge University Press Warehouse, Ave Maria Lane",
 year = "1890"
}
+\begin{chunk}{ignore}
+\bibitem[Dongarra 88a]{REFDON88a} Dongarra, Jack J.; Du Croz, Jeremy;
+Hammarling, Sven; Hanson, Richard J.
+``ALGORITHM 656: An Extended Set of Basic Linear Algebra Subprograms:
+Model Implementation and Test Programs''
+ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988,
+pp 1832
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Taivalsaari 96]{Tai96} Taivalsaari, Antero
``On the Notion of Inheritance''
ACM Computing Surveys, Vol 28 No 3 Sept 1996 pp438479
+\bibitem[Dongarra 90]{REFDON90} Dongarra, Jack J.; Du Croz, Jeremy;
+Hammarling, Sven; Duff, Iain S.
+``A Set of Level 3 Basic Linear Algebra Subprograms''
+ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990,
+pp 117
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Temme 87]{Tem87} Temme N M.
``On the Computation of the Incomplete Gamma Functions for Large Values of
the Parameters''
Algorithms for Approximation. (ed J C Mason and M G Cox)
Oxford University Press. (1987)
+\bibitem[Dongarra 90a]{REFDON90a} Dongarra, Jack J.; Du Croz, Jeremy;
+Hammarling, Sven; Duff, Iain S.
+``ALGORITHM 679: A Set of Level 3 Basic Linear Algebra Subprograms:
+Model Implementation and Test Programs''
+ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990,
+pp 1828
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Temperton 83a]{Tem83a} Temperton C.
``Selfsorting Mixedradix Fast Fourier Transforms''
J. Comput. Phys. 52 123. (1983)
+\bibitem[Ducos 00]{Duc00} Ducos, Lionel
+``Optimizations of the subresultant algorithm''
+Journal of Pure and Applied Algebra V145 No 2 Jan 2000 pp149163
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Temperton 83b]{Tem83b} Temperton C.
``Fast MixedRadix Real Fourier Transforms''
J. Comput. Phys. 52 340350. (1983)
+\bibitem[Duff 77]{Duff77} Duff I S,
+``MA28  a set of Fortran subroutines for sparse unsymmetric linear
+equations''
+A.E.R.E. Report R.8730. HMSO. (1977)
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Thur94,
 author = "Thurston, William P.",
 title = "On Proof and Progress in Mathematics",
 journal = "Bulletin AMS",
 volume = "30",
 number = "2",
 month = "April",
 year = "1994",
 url = "http://www.ams.org/journals/bull/19943002/S027309791994005026/S027309791994005026.pdf",
 paper = "Thur94.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Duval 96a]{Duva96a} Duval, D.; Gonz\'alezVega, L.
+``Dynamic Evaluation and Real Closure''
+Mathematics and Computers in Simulation 42 pp 551560 (1996)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva96a.pdf
+ abstract = "
+ The aim of this paper is to present how the dynamic evaluation method
+ can be used to deal with the real closure of an ordered field. Two
+ kinds of questions, or tests, may be asked in an ordered field:
+ equality tests $(a=b?)$ and sign tests $(a > b?)$. Equality tests are
+ handled through splittings, exactly as in the algebraic closure of a
+ field. Sign tests are handled throug a structure called ``Tarski data
+ type''."
\end{chunk}
\subsection{U} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Unknown 61]{Unk61} Unknown
``Chebyshevseries''
Modern Computing Methods
Chapter 8. NPL Notes on Applied Science (2nd Edition). 16 HMSO. 1961
+\bibitem[Duval 96]{Duva96} Duval, D.; Reynaud, J.C.
+``Sketches and Computations over Fields''
+Mathematics and Computers in Simulation 42 pp 363373 (1996)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva96.pdf
+ abstract = "
+ The goal of this short paper is to describe one possible use of
+ sketches in computer algebra. We show that sketches are a powerful
+ tool for the description of mathematical structures and for the
+ description of computations."
\end{chunk}
\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Van Dooren 76]{vDDR76} Van Dooren P.; De Ridder L.
``An Adaptive Algorithm for Numerical Integration over an Ndimensional
Cube''
J. Comput. Appl. Math. 2 207217. (1976)
+\bibitem[Duval 94a]{Duva94a} Duval, D.; Reynaud, J.C.
+``Sketches and Computation (Part I): Basic Definitions and Static Evaluation''
+Mathematical Structures in Computer Science, 4, p 185238 Cambridge University Press (1994)
+\verbjournals.cambridge.org/abstract_S0960129500000438
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva94a.pdf
+ abstract = "
+ We define a categorical framework, based on the notion of {\sl
+ sketch}, for specification and evaluation in the sense of algebraic
+ specifications and algebraic programming. This framework goes far
+ beyond our initial motivations, which was to specify computation with
+ algebraic numbers. We begin by redefining sketches in order to deal
+ explicitly with programs. Expressions and terms are carefully defined
+ and studied, then {\sl quasiprojective sketches} are introduced. We
+ describe {\sl static evaluation} in these sketches: we propose a
+ rigorous basis for evaluation in the corresponding structures. These
+ structures admit an initial model, but are not necessarily
+ equational. In Part II (Duval and Reynaud 1994), we study a more
+ general process, called {\sl dynamic evaluation}, for structures that
+ may have no initial model."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[van Hoeij 94]{REFvH94} van Hoeij, M.
``An algorithm for computing an integral
basis in an algebraic function field''
{\sl J. Symbolic Computation}
18(4):353364, October 1994
+\bibitem[Duval 94b]{Duva94b} Duval, D.; Reynaud, J.C.
+``Sketches and Computation (Part II): Dynamic Evaluation and Applications''
+Mathematical Structures in Computer Science, 4, p 239271. Cambridge University Press (1994)
+\verbjournals.cambridge.org/abstract_S096012950000044X
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva94b.pdf
+ abstract = "
+ In the first part of this paper (Duval and Reynaud 1994), we defined a
+ categorical framework, based on the notion of {\sl sketch}, for
+ specification and evaluation in the senses of algebraic specification
+ and algebraic programming. {\sl Static evaluation} in {\sl
+ quasiprojective sketches} was defined in Part I; in this paper, {\sl
+ dynamic evaluation} is introduced. It deals with more general
+ structures, which may have no initial model. Until now, this process
+ has not been used in algebraic specification systems, but computer
+ algebra systems are beginning to use it as a basic tool. Finally, we
+ give some applications of dynamic evaluation to computation in field
+ extensions."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Van Loan 92]{Van92} Van Loan, C.
``Computational Frameworks for the Fast Fourier Transform''
SIAM Philadelphia. (1992)
+\bibitem[Duval 94c]{Duva94c} Duval, Dominique
+``Algebraic Numbers: An Example of Dynamic Evaluation''
+J. Symbolic Computation 18, 429445 (1994)
+\verbwww.sciencedirect.com/science/article/pii/S0747717106000551
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Duva94c.pdf
+ abstract = "
+ Dynamic evaluation is presented through examples: computations
+ involving algebraic numbers, automatic case discussion according to
+ the characteristic of a field. Implementation questions are addressed
+ too. Finally, branches are presented as ``dual'' to binary functions,
+ according to the approach of sketch theory."
\end{chunk}
\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Wait 85]{WM85} Wait R.; Mitchell A R.
``Finite Element Analysis and Application''
Wiley. (1985)
+\bibitem[Fateman 08]{Fat08} Fateman, Richard
+``Revisiting numeric/symbolic indefinite integration of rational functions, and extensions''
+\verbwww.eecs.berkeley.edu/~fateman/papers/integ.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat08.pdf
+ abstract = "
+ We know we can solve this problem: Given any rational function
+ $f(x)=p(x)/q(x)$, where $p$ and $q$ are univariate polynomials over
+ the rationals, compute its {\sl indefinite} integral, using if
+ necessary, algebraic numbers. But in many circumstances an approximate
+ result is more likely to be of use. Furthermore, it is plausible that
+ it would be more useful to solve the problem to allow definite
+ integration, or introduce additional parameters so that we can solve
+ multiple definite integrations. How can a computer algebra system
+ best answer the more useful questions? Finally, what if the integrand
+ is not a ratio of polynomials, but something more challenging?"
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wang 92]{Wang92} Wang, D.M.
``An implementation of the characteristic set method in Maple''
Proc. DISCO'92 Bath, England
+\begin{chunk}{axiom.bib}
+@misc{Flet01,
+ author = "Fletcher, John P.",
+ title = "Symbolic processing of Clifford Numbers in C++",
+ year = "2001",
+ journal = "Paper 25, AGACSE 2001."
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@misc{Flet09,
+ author = "Fletcher, John P.",
+ title = "Clifford Numbers and their inverses calculated using the matrix
+ representation",
+ publisher = "Chemical Engineering and Applied Chemistry, School of
+ Engineering and Applied Science, Aston University, Aston Triangle,
+ Birmingham B4 7 ET, U. K.",
+ url =
+ "http://www.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ward 75]{War75} Ward, R C.
``The Combination Shift QZ Algorithm''
SIAM J. Numer. Anal. 12 835853. 1975
+\bibitem[Fletcher 81]{Fle81} Fletcher R
+``Practical Methods of Optimization''
+Vol 2. Constrained Optimization. Wiley. (1981)
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Watt03,
 author = "Watt, Stephen",
 title = "Aldor",
 url = "http://www.aldor.org",
 year = "2003"
+@article{Floy63,
+ author = "Floyd, R. W.",
+ title = "Semantic Analysis and Operator Precedence",
+ journal = "JACM",
+ volume = "10",
+ number = "3",
+ pages = "316333",
+ year = "1963"
}
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Weil71,
 author = "Weil, Andr\'{e}",
 title = "Courbes alg\'{e}briques et vari\'{e}t\'{e}s Abeliennes",
 year = "1971"
}
+\begin{chunk}{ignore}
+\bibitem[Forsythe 57]{For57} Forsythe G E,
+``Generation and use of orthogonal polynomials for data fitting
+with a digital computer''
+J. Soc. Indust. Appl. Math. 5 7488. (1957)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Weisstein]{Wein} Weisstein, Eric W.
``Hypergeometric Function''
MathWorld  A Wolfram Web Resource
\verbmathworld.wolfram.com/HypergeometricFunction.html
+\bibitem[Fortenbacher 90]{REFFor90} Fortenbacher, A.
+``Efficient type inference and coercion in computer algebra''
+Design and Implementation of Symbolic Computation Systems (DISCO 90)
+A. Miola, (ed) vol 429 of Lecture Notes in Computer Science
+SpringerVerlag, pp5660
+ abstract = "
+ Computer algebra systems of the new generation, like Scratchpad, are
+ characterized by a very rich type concept, which models the
+ relationship between mathematical domains of computation. To use these
+ systems interactively, however, the user should be freed of type
+ information. A type inference mechanism determines the appropriate
+ function to call. All known models which allow to define a semantics
+ for type inference cannot express the rich ``mathematical'' type
+ structure, so presently type inference is done heuristically. The
+ following paper defines a semantics for a subproblem thereof, namely
+ coercion, which is based on rewrite rules. From this definition, and
+ efficient coercion algorith for Scratchpad is constructed using graph
+ techniques."
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Weit03,
 author = "Weitz, E.",
 title = "CLWHO Yet another Lisp markup language",
 year = "2003",
 url = "http://www.weitz.de/clwho/"
}
+\begin{chunk}{ignore}
+\bibitem[Fox 68]{Fox68} Fox L.; Parker I B.
+``Chebyshev Polynomials in Numerical Analysis''
+Oxford University Press. (1968)
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Weit06,
 author = "Weitz, E.",
 title = "HUNCHENTOOT  The Common Lisp web server formerly known as TBNL",
 year = "2006",
 url = "http://www.weitz.de/hunchentoot"
}
+\begin{chunk}{ignore}
+\bibitem[Franke 80]{FN80} Franke R.; Nielson G
+``Smooth Interpolation of Large Sets of Scattered Data''
+Internat. J. Num. Methods Engrg. 15 16911704. (1980)
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Fritsch 82]{Fri82} Fritsch F N
+``PCHIP Final Specifications''
+Report UCID30194. Lawrence Livermore National Laboratory. (1982)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wesseling 82a]{Wes82a} Wesseling, P.
``MGD1  A Robust and Efficient Multigrid Method''
Multigrid Methods. Lecture Notes in Mathematics. 960
SpringerVerlag. 614630. (1982)
+\bibitem[Fritsch 84]{FB84} Fritsch F N.; Butland J.
+``A Method for Constructing Local Monotone Piecewise Cubic Interpolants''
+SIAM J. Sci. Statist. Comput. 5 300304. (1984)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wesseling 82b]{Wes82b} Wesseling, P.
``Theoretical Aspects of a Multigrid Method''
SIAM J. Sci. Statist. Comput. 3 387407. (1982)
+\bibitem[Froberg 65]{Fro65} Froberg C E.
+``Introduction to Numerical Analysis''
+AddisonWesley. 181187. (1965)
\end{chunk}
+\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Wicks 89]{Wic89} Wicks, Mark; Carlisle, David, Rahtz, Sebastian
``dvipdfm.def''
\verbweb.mit.edu/texsrc/source/latex/graphics/dvipdfm.def
+\bibitem[Garcia 95]{Ga95} Garcia, A.; Stichtenoth, H.
+``A tower of ArtinSchreier extensions of function fields attaining the
+DrinfeldVladut bound''
+Invent. Math., vol. 121, 1995, pp. 211222.
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wiki 3]{Wiki3}.
``Givens Rotations''
\verben.wikipedia.org/wiki/Givens_rotation
+\bibitem[Gathen 90a]{Gat90a} Gathen, Joachim von zur; Giesbrecht, Mark
+``Constructing Normal Bases in Finite Fields''
+J. Symbolic Computation pp 547570 (1990)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Gat90a.pdf
+ abstract = "
+ An efficient probabilistic algorithm to find a normal basis in a
+ finite field is presented. It can, in fact, find an element of
+ arbitrary prescribed additive order. It is based on a density estimate
+ for normal elements. A similar estimate yields a probabilistic
+ polynomialtime reduction from finding primitive normal elements to
+ finding primitive elements."
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Wiki14a,
 author = "ProofWiki",
 title = "Euclidean Algorithm",
 url = "http://proofwiki.org/wiki/Euclidean_Algorithm"
}
+\begin{chunk}{ignore}
+\bibitem[Gathen 90]{Gat90} Gathen, Joachim von zur
+``Functional Decomposition Polynomials: the Tame Case''
+Journal of Symbolic Computation (1990) 9, 281299
\end{chunk}
\begin{chunk}{axiom.bib}
@misc{Wiki14b,
 author = "ProofWiki",
 title = "Division Theorem",
 url = "http://proofwiki.org/wiki/Division_Theorem"
+@book{Gath99,
+ author = {{von zur Gathen}, Joachim and Gerhard, J\"urgen},
+ title = "Modern Computer Algebra",
+ publisher = "Cambridge University Press",
+ year = "1999",
+ isbn = "0521641764"
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Williamson 85]{Wil85} Williamson, S.G.
``Combinatorics for Computer Science''
Computer Science Press, 1985.
+\bibitem[Gautschi 79a]{Gau79a} Gautschi W.
+``A Computational Procedure for Incomplete Gamma Functions''
+ACM Trans. Math. Softw. 5 466481. (1979)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wilkinson 71]{WR71} Wilkinson J H.; Reinsch C.
``Handbook for Automatic Computation II, Linear Algebra''
SpringerVerlag. 1971
+\bibitem[Gautschi 79b]{Gau79b} Gautschi W.
+``Algorithm 542: Incomplete Gamma Functions''
+ACM Trans. Math. Softw. 5 482489. (1979)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wilkinson 63]{Wil63} Wilkinson J H.
``Rounding Errors in Algebraic Processes''
 Chapter 2. HMSO. (1963)
+\bibitem[Gentlemen 69]{Gen69} Gentlemen W M
+``An Error Analysis of Goertzel's (Watt's) Method for Computing
+Fourier Coefficients''
+Comput. J. 12 160165. (1969)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wilkinson 65]{Wil65} Wilkinson J H.
``The Algebraic Eigenvalue Problem''
 Oxford University Press. (1965)
+\bibitem[Gentleman 73]{Gen73} Gentleman W M.
+``Leastsquares Computations by Givens Transformations without Square Roots''
+J. Inst. Math. Applic. 12 329336. (1973)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wilkinson 78]{Wil78} Wilkinson J H.
``Singular Value Decomposition  Basic Aspects''
Numerical Software  Needs and Availability.
(ed D A H Jacobs) Academic Press. (1978)
+\bibitem[Gentleman 74]{Gen74} Gentleman W M.
+``Algorithm AS 75. Basic Procedures for Large Sparse or
+Weighted Linear Leastsquares Problems''
+Appl. Statist. 23 448454. (1974)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wilkinson 79]{Wil79} Wilkinson J H.
``Kronecker's Canonical Form and the QZ Algorithm''
Linear Algebra and Appl. 28 285303. 1979
+\bibitem[Gentlemen 74a]{GM74a} Gentleman W. M.; Marovich S. B.
+``More on algorithms that reveal properties of floating point
+arithmetic units''
+Comms. of the ACM, 17, 276277. (1974)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wisbauer 91]{Wis91} Wisbauer, R.
``Bimodule Structure of Algebra''
Lecture Notes Univ. Duesseldorf 1991
+\bibitem[Genz 80]{GM80} Genz A C.; Malik A A.
+``An Adaptive Algorithm for Numerical Integration over an Ndimensional
+Rectangular Region''
+J. Comput. Appl. Math. 6 295302. (1980)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[WoerzBusekros 80]{Woe80} WoerzBusekros, A.
``Algebra in Genetics''
Lectures Notes in Biomathematics 36, SpringerVerlag, Heidelberg, 1980
+\bibitem[Gill 72]{GM72} Gill P E.; Miller G F.
+``An Algorithm for the Integration of Unequally Spaced Data''
+Comput. J. 15 8083. (1972)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wolberg 67]{Wol67} Wolberg J R.
``Prediction Analysis''
Van Nostrand. (1967)
+\bibitem[Gill 74b]{GM74b} Gill P E.; Murray W. (eds)
+``Numerical Methods for Constrained Optimization''
+Academic Press. (1974)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wolfram 09]{Wo09} Wolfram Research
\verbmathworld.wolfram.com/Quaternion.html
+\bibitem[Gill 76a]{GM76a} Gill P E.; Murray W.
+``Minimization subject to bounds on the variables''
+Report NAC 72. National Physical Laboratory. (1976)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wu 87]{WU87} Wu, W.T.
``A Zero Structure Theorem for polynomial equations solving''
MM Research Preprints, 1987
+\bibitem[Gill 76b]{GM76b} Gill P E.; Murray W.
+``Algorithms for the Solution of the Nonlinear Leastsquares Problem''
+NAC 71 National Physical Laboratory. (1976)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wynn 56]{Wynn56} Wynn P.
``On a Device for Computing the $e_m(S_n )$ Transformation''
Math. Tables Aids Comput. 10 9196. (1956)
+\bibitem[Gill 78]{GM78} Gill P E.; Murray W.
+``Algorithms for the Solution of the Nonlinear Leastsquares Problem''
+SIAM J. Numer. Anal. 15 977992. (1978)
\end{chunk}
\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Zakrajsek 02]{Zak02} Zakrajsek, Helena
``Applications of Hermite transform in computer algebra''
\verbwww.imfm.si/preprinti/PDF/00835.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Zak02.pdf
+\bibitem[Gill 79]{GM79} Gill P E.; Murray W;
+``Conjugategradient Methods for Largescale Nonlinear Optimization''
+Technical Report SOL 7915. Department of Operations Research,
+Stanford University. (1979)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
let $L$ be a linear differential operator with polynomial coefficients.
We show that there is an isomorphism of differential operators
${\bf D_\alpha}$ and an integral transform ${\bf H_\alpha}$ (called the
Hermite transform) on functions for which $({\bf D_\alpha}{\bf L})f(x)=0$
implies ${\bf L}{\bf H_alpha}(f)(x)=0$. We present an algorithm that
computes the Hermite transform of a rational function and use it to find
$n+1$ linearly independent solutions of ${\bf L}y=0$ when
$({\bf D_\alpha}{\bf L})f(x)=0$ has a rational solution with $n$
distinct finite poles.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@misc{Zdan14,
 author = "Zdancewic, Steve and Martin, Milo M.K.",
 title = "Vellvm: Verifying the LLVM",
 url = "http://www.cis.upenn.edu/~stevez/vellvm"
}
+\begin{chunk}{ignore}
+\bibitem[Gill 81]{GMW81} Gill P E.; Murray W.; Wright M H.
+``Practical Optimization''
+Academic Press. 1981
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Zhi 97]{Zhi97} Zhi, Lihong
``Optimal Algorithm for Algebraic Factoring''
\verbwww.mmrc.iss.ac.cn/~lzhi/Publications/zopfac.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Zhi97.pdf
+\bibitem[Gill 82]{GMW82} Gill P E.; Murray W.; Saunders M A.; Wright M H.
+``The design and implementation of a quadratic programming algorithm''
+Report SOL 827. Department of Operations Research,
+Stanford University. (1982)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper presents an optimized method for factoring multivariate
polynomials over algebraic extension fields which defined by an
irreducible ascending set. The basic idea is to convert multivariate
polynomials to univariate polynomials and algebraic extensions fields
to algebraic number fields by suitable integer substitutions, then
factorize the univariate polynomials over the algebraic number fields.
Finally, construct multivariate factors of the original polynomial by
Hensel lemma and TRUEFACTOR test. Some examples with timing are
included.
\end{adjustwidth}

\section{Special Topics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Solving Systems of Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Bronstein 86]{Bro86} Bronstein, Manuel
``Gsolve: a faster algorithm for solving systems of algebraic equations''
Proc of 5th ACM SYMSAC (1986) pp247249 ISBN 0897911997
+\bibitem[Gill 84a]{GMSW84a} Gill P E.; Murray W.; Saunders M A.; Wright M H
+``User's Guide for SOL/QPSOL Version 3.2''
+Report SOL 845. Department of Operations Research, Stanford University. 1984
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We apply the elimination property of Gr\"obner bases with respect to
pure lexicographic ordering to solve systems of algebraic equations.
We suggest reasons for this approach to be faster than the resultant
technique, and give examples and timings that show that it is indeed
faster and more correct, than MACSYMA's solve.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Gill 84b]{GMSW84b} Gill P E.; Murray W.; Saunders M A.; Wright M H
+``Procedures for Optimization Problems with a Mixture of
+Bounds and General Linear Constraints''
+ACM Trans. Math. Softw. 10 282298. 1984
\subsection{Numerical Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 99]{Bro99} Bronstein, Manuel
``Fast Deterministic Computation of Determinants of Dense Matrices''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro99.pdf
+\bibitem[Gill 86a]{GMSW86a} Gill P E.; Hammarling S.; Murray W.;
+Saunders M A.; Wright M H.
+``User's Guide for LSSOL (Version 1.0)''
+Report SOL 861. Department of Operations Research, Stanford University. 1986
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this paper we consider deterministic computation of the exact
determinant of a dense matrix $M$ of integers. We present a new
algorithm with worst case complexity
\[O(n^4(log n+ log \verb?M?)+x^3 log^2 \verb?M?)\],
where $n$ is the dimension of the matrix
and \verb?M? is a bound on the entries in $M$, but with
average expected complexity
\[O(n^4+m^3(log n + log \verb?M?)^2)\],
assuming some plausible properties about the distribution of $M$.
We will also describe a practical version of the algorithm and include
timing data to compare this algorithm with existing ones. Our result
does not depend on ``fast'' integer or matrix techniques.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Kelsey 00]{Kel00} Kelsey, Tom
``Exact Numerical Computation via Symbolic Computation''
\verbtom.host.cs.standrews.ac.uk/pub/ccapaper.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Kel00.pdf
+\bibitem[Gill 86b]{GMSW86b} Gill P E.; Murray W.; Saunders M A.; Wright M H.
+``Some Theoretical Properties of an Augmented Lagrangian Merit Function''
+Report SOL 866R. Department of Operations Research, Stanford University. 1986
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We provide a method for converting any symbolic algebraic expression
that can be converted into a floating point number into an exact
numeric representation. We use this method to demonstrate a suite of
procedures for the representation of, and arithmetic over, exact real
numbers in the Maple computer algebra system. Exact reals are
represented by potentially infinite lists of binary digits, and
interpreted as sums of negative powers of the golden ratio.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Yang 14]{Yang14} Yang, Xiang; Mittal, Rajat
``Acceleration of the Jacobi iterative method by factors exceeding 100
using scheduled relation''
\verbengineering.jhu.edu/fsag/wpcontent/uploads/sites/23/2013/10
\verbJCP_revised_WebPost.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Yang14.pdf
+\bibitem[Gladwell 79]{Gla79} Gladwell I
+``Initial Value Routines in the NAG Library''
+ACM Trans Math Softw. 5 386400. (1979)
\end{chunk}
\subsection{Special Functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Corless 05]{Corl05} Corless, Robert M.; Jeffrey, David J.;
Watt, Stephen M.; Bradford, Russell; Davenport, James H.
``Reasoning about the elementary functions of complex analysis''
\verbwww.csd.uwo.ca/~watt/pub/reprints/2002amaireasoning.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Corl05.pdf
+\bibitem[Gladwell 80]{GS80} Gladwell I.; Sayers D K
+``Computational Techniques for Ordinary Differential Equations''
+Academic Press. 1980
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Ng 68]{Ng68} Ng, Edward W.; Geller, Murray
``A Table of Integrals of the Error functions''
\verbnvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Ng68.pdf
+\bibitem[Gladwell 86]{Gla86} Gladwell I
+``Vectorisation of one dimensional quadrature codes''
+Techincal Report. TR7/86 NAG. (1986)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This is a compendium of indefinite and definite integrals of products
of the Error functions with elementary and transcendental functions.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Gladwell 87]{Gla87} Gladwell I
+``The NAG Library Boundary Value Codes''
+Numerical Analysis Report. 134 Manchester University. (1987)
\subsubsection{Exponential Integral $E_1(x)$} %%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Geller 69]{Gell69} Geller, Murray; Ng, Edward W.
``A Table of Integrals of the Exponential Integral''
\verbnvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Gell69.pdf
+\bibitem[Goedel 40]{God40} Goedel
+``The consistency of the continuum hypothesis''
+Ann. Math. Studies, Princeton Univ. Press, 1940
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This is a compendium of indefinite and definite integrals of products
of the Exponential Integral with elementary or transcendental functions.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@techreport{Segl98,
 author = "Segletes, S.B.",
 title = "A compact analytical fit to the exponential integral $E_1(x)$",
 year = "1998",
 institution = "U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD",
 type = "Technical Report",
 number = "ARLTR1758",
 paper = "Segl98.pdf"
}

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
A fourparameter fit is developed for the class of integrals known as
the exponential integral (real branch). Unlike other fits that are
piecewise in nature, the current fit to the exponential integral is
valid over the complete domain of the function (compact) and is
everywhere accurate to within $\pm 0.0052\%$ when evaluating the first
exponential integral, $E_1$. To achieve this result, a methodology
that makes use of analytically known limiting behaviors at either
extreme of the domain is employed. Because the fit accurately captures
limiting behaviors of the $E_1$ function, more accuracy is retained
when the fit is used as part of the scheme to evaluate higherorder
exponential integrals, $E_n$, as compared with the use of bruteforce
fits to $E_1$, which fail to accurately model limiting
behaviors. Furthermore, because the fit is compact, no special
accommodations are required (as in the case of spliced piecewise fits)
to smooth the value, slope, and higher derivatives in the transition
region between two piecewise domains. The general methodology employed
to develop this fit is outlined, since it may be used for other
problems as well.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Segletes 09]{Se09} Segletes, S.B.
``Improved fits for $E_1(x)$ {\sl vis\'avis} those presented in ARLTR1758
Technical Report ARLTR1758, U.S. Army Ballistic Research Laboratory,
Aberdeen Proving Ground, MD, September 1998
%\verbaxiomdeveloper.org/axiomwebsite/papers/Se09.pdf

\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
This is a writeup detailing the more accurate fits to $E_1(x)$,
relative to those presented in ARLTR1758. My actual fits are to
\[F1 =[x\ exp(x) E_1(x)]\] which spans a functional range from 0 to 1.
The best accuracy I have been yet able to achieve, defined by limiting
the value of \[[(F1)_{fit}  F1]/F1\] over the domain, is approximately
3.1E07 with a 12parameter fit, which unfortunately isn't quite to 32bit
floatingpoint accuracy. Nonetheless, the fit is not a piecewise fit,
but rather a single continuous function over the domain of nonnegative x,
which avoids some of the problems associated with piecewise domain splicing.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Goldman 87]{Gold87} Goldman, L.
+``Integrals of multinomial systems of ordinary differential equations''
+J. of Pure and Applied Algebra, 45, 225240 (1987)
+\verbwww.sciencedirect.com/science/article/pii/0022404987900727/pdf
+\verb?md5=5a0c70643eab514ccf47d80e4fc6ec5a&
+\verbpid=1s2.00022404987900727main.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Gold87.pdf
\subsection{Polynomial GCD} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Knuth 71]{STPGCDKnu71} Knuth, Donald
``The Art of Computer Programming''
2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing,
AddisonWesley 1971, section 4.6 pp399505
+\bibitem[Gollan 90]{GG90} H. Gollan; J. Grabmeier
+``Algorithms in Representation Theory and
+their Realization in the Computer Algebra System Scratchpad''
+Bayreuther Mathematische Schriften, Heft 33, 1990, 123
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ma 90]{STPGCDMa90} Ma, Keju; Gathen, Joachim von zur
``Analysis of Euclidean Algorithms for Polynomials over Finite Fields''
J. Symbolic Computation (1990) Vol 9 pp429455\hfill{}
\verbwww.researchgate.net/publication/220161718_Analysis_of_Euclidean_
\verbAlgorithms_for_Polynomials_over_Finite_Fields/file/
\verb60b7d52b326a1058e4.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/STPGCDMa90.pdf
+\bibitem[Golub 89]{GL89} Golub, Gene H.; Van Loan, Charles F.
+``Matrix Computations''
+Johns Hopkins University Press ISBN 0801837723 (1989)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper analyzes the Euclidean algorithm and some variants of it
for computing the greatest common divisor of two univariate polynomials
over a finite field. The minimum, maximum, and average number of
arithmetic operations both on polynomials and in the ground field
are derived.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Naylor 00a]{N00} Naylor, Bill
``Polynomial GCD Using Straight Line Program Representation''
PhD. Thesis, University of Bath, 2000
\verbwww.sci.csd.uwo.ca/~bill/thesis.ps
%\verbaxiomdeveloper.org/axiomwebsite/papers/N00.pdf
+\bibitem[Golub 96]{GL96} Golub, Gene H.; Van Loan, Charles F.
+``Matrix Computations''
+Johns Hopkins University Press ISBN 9780801854149 (1996)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This thesis is concerned with calculating polynomial greatest common
divisors using straight line program representation.

In the Introduction chapter, we introduce the problem and describe
some of the traditional representations for polynomials, we then talk
about some of the general subjects central to the thesis, terminating
with a synopsis of the category theory which is central to the Axiom
computer algebra system used during this research.

The second chapter is devoted to describing category theory. We follow
with a chapter detailing the important sections of computer code
written in order to investigate the straight line program subject.
The following chapter on evalution strategies and algorithms which are
dependant on these follows, the major algorith which is dependant on
evaluation and which is central to our theis being that of equality
checking. This is indeed central to many mathematical problems.
Interpolation, that is the determination of coefficients of a
polynomial is the subject of the next chapter. This is very important
for many straight line program algorithms, as their noncanonical
structure implies that it is relatively difficult to determine
coefficients, these being the basic objects that many algorithms work
on. We talk about three separate interpolation techniques and compare
their advantages and disadvantages. The final two chapters describe
some of the results we have obtained from this research and finally
conclusions we have drawn as to the viability of the straight line
program approach and possible extensions.

Finally we terminate with a number of appendices discussing side
subjects encountered during the thesis.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Shoup 93]{STPGCDSh93} Shoup, Victor
``Factoring Polynomials over Finite Fields: Asymptotic Complexity vs
Reality*''
Proc. IMACS Symposium, Lille, France, (1993)
\verbwww.shoup.net/papers/lille.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/STPGCDSh93.pdf
+\bibitem[Grabmeier]{Grab} Grabmeier, J.
+``On Plesken's root finding algorithm''
+in preparation
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper compares the algorithms by Berlekamp, Cantor and Zassenhaus,
and Gathen and Shoup to conclude that
(a) if large polynomials are factored the FFT should be used for polynomial
multiplication and division,
(b) Gathen and Shoup should be used if the number of irreducible factors
of $f$ is small.
(c) if nothing is know about the degrees of the factors then Berlekamp's
algorithm should be used
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Gathen 01]{STPGCDGa01} Gathen, Joachim von zur; Panario, Daniel
``Factoring Polynomials Over Finite Fields: A Survey''
J. Symbolic Computation (2001) Vol 31, pp317\hfill{}
\verbpeople.csail.mit.edu/dmoshdov/courses/codes/polyfactorization.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/STPGCDGa01.pdf
 keywords = "survey",
+\bibitem[Grebmeier 87]{GK87} Grabmeier, J.; Kerber, A.;
+``The Evaluation of Irreducible Polynomial Representations of the General
+Linear Groups and of the Unitary Groups over Fields of Characteristic 0''
+Acta Appl. Math. 8 (1987), 271291
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This survey reviews several algorithms for the factorization of
univariate polynomials over finite fields. We emphasize the main ideas
of the methods and provide and uptodate bibliography of the problem.
This paper gives algorithms for {\sl squarefree factorization},
{\sl distinctdegree factorization}, and {\sl equaldegree factorization}.
The first and second algorithms are deterministic, the third is
probabilistic.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[van Hoeij]{Hoeij04} Hoeij, Mark van; Monagen, Michael
``Algorithms for Polynomial GCD Computation over Algebraic Function Fields''
\verbwww.cecm.sfu.ca/personal/mmonagan/papers/AFGCD.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Hoeij04.pdf
+\bibitem[Grabmeier 92]{REFGS92} Grabmeier, J.; Scheerhorn, A.
+``Finite fields in Axiom''
+AXIOM Technical Report TR7/92 (ATR/5)(NP2522),
+Numerical Algorithms Group, Inc., Downer's
+Grove, IL, USA and Oxford, UK, 1992
+\verbwww.nag.co.uk/doc/TechRep/axiomtr.html
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Let $L$ be an algebraic function field in $k \ge 0$ parameters
$t_1,\ldots,t)k$. Let $f_1$, $f_2$ be nonzero polynomials in
$L[x]$. We give two algorithms for computing their gcd. The first, a
modular GCD algorithm, is an extension of the modular GCD algorithm
for Brown for {\bf Z}$[x_1,\ldots,x_n]$ and Encarnacion for {\bf
Q}$(\alpha[x])$ to function fields. The second, a fractionfree
algorithm, is a modification of the Moreno Maza and Rioboo algorithm
for computing gcds over triangular sets. The modification reduces
coefficient grownth in $L$ to be linear. We give an empirical
comparison of the two algorithms using implementations in Maple.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Wang 78]{Wang78} Wang, Paul S.
``An Improved Multivariate Polynomial Factoring Algorithm''
Mathematics of Computation, Vol 32, No 144 Oct 1978, pp12151231
\verbwww.ams.org/journals/mcom/197832144/S00255718197805682843/
\verbS00255718197805682843.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Wang78.pdf
+\bibitem[Granville 1911]{Gran1911} Granville, William Anthony
+``Elements of the Differential and Integral Calculus''
+\verbdjm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Gran1911.pdf
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A new algorithm for factoring multivariate polynomials over the
integers based on an algorithm by Wang and Rothschild is described.
The new algorithm has improved strategies for dealing with the known
problems of the original algorithm, namely, the leading coefficient
problem, the badzero problem and the occurence of extraneous factors.
It has an algorithm for correctly predetermining leading coefficients
of the factors. A new and efficient padic algorith named EEZ is
described. Basically it is a linearly convergent variablebyvariable
parallel construction. The improved algorithm is generally faster and
requires less store than the original algorithm. Machine examples with
comparative timing are included.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Wiki 4]{Wiki4}.
``Polynomial greatest common divisor''
\verben.wikipedia.org/wiki/Polynomial_greatest_common_divisor
+\bibitem[Gruntz 93]{Gru93} Gruntz, Dominik
+``Limit computation in computer algebra''
+\verbalgo.inria.fr/seminars/sem9293/gruntz.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Gru93.pdf
+ abstract = "
+ The automatic computation of limits can be reduced to two main
+ subproblems. The first one is asymptotic comparison where one must
+ decide automatically which one of two functions in a specified class
+ dominates the other one asymptotically. The second one is asymptotic
+ cancellation and is usually exemplified by
+ \[e^x[exp(1/x+e^{x})exp(1/x)],\quad{}x \leftarrow \infty\]
+
+ In this example, if the sum is expanded in powers of $1/x$, the
+ expansion always yields $O(x^{k})$, and this is not enough to
+ conclude.
+
+ In 1990, J.Shackell found an algorithm that solved both these problems
+ for the case of $explog$ functions, i.e. functions build by recursive
+ application of exponential, logarithm, algebraic extension and field
+ operations to one variable and the rational numbers. D. Gruntz and
+ G. Gonnet propose a slightly different algorithm for explog
+ functions. Extensions to larger classes of functions are also
+ discussed."
\end{chunk}
\subsection{Category Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Baez 09]{Baez09} Baez, John C.; Stay, Mike
``Physics, Topology, Logic and Computation: A Rosetta Stone''
\verbarxiv.org/pdf/0903.0340v3.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Baez09.pdf
+\begin{chunk}{axiom.bib}
+@article{Hach95,
+ author = "Hach\'e, G. and Le Brigand, D.",
+ title = "Effective construction of algebraic geometry codes",
+ journal = "IEEE Transaction on Information Theory",
+ volume = "41",
+ month = "November",
+ year = "1995",
+ pages = "16151628"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In physics, Feynman diagrams are used to reason about quantum
processes. In the 1980s, it became clear that underlying these
diagrams is a powerful analogy between quantum physics and
topology. Namely, a linear operator behaves very much like a
``cobordism'': a manifold representing spacetime, going between two
manifolds representing space. But this was just the beginning: simiar
diagrams can be used to reason about logic, where they represent
proofs, and computation, where they represent programs. With the rise
of interest in quantum cryptography and quantum computation, it became
clear that there is an extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make
some of these analogies precise using the concept of ``closed
symmetric monodial category''. We assume no prior knowledge of
category theory, proof theory or computer science.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Meijer 91]{Meij91} Meijer, Erik; Fokkinga, Maarten; Paterson, Ross
``Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire''
\verbeprints.eemcs.utwente.nl/7281/01/dbutwente40501F46.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Meij91.pdf
+\begin{chunk}{axiom.bib}
+@article{Hach95a,
+ author = "Hach\'e, G.",
+ title = "Computation in algebraic function fields for effective
+ construction of algebraicgeometric codes",
+ journal = "Lecture Notes in Computer Science",
+ volume = "948",
+ year = "1995",
+ pages = "262278"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}
+\begin{chunk}{axiom.bib}
+@phdthesis{Hach96,
+ author = "Hach\'e, G.",
+ title = "Construction effective des codes g\'eom\'etriques",
+ school = "l'Universit\'e Pierre et Marie Curie (Paris 6)",
+ year = "1996",
+ month = "Septembre"
+}
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Youssef 04]{You04} Youssef, Saul
``Prospects for Category Theory in Aldor''
October 2004
%\verbaxiomdeveloper.org/axiomwebsite/papers/You04.pdf
+\bibitem[Hall 76]{HW76} Hall G.; Watt J M. (eds),
+``Modern Numerical Methods for Ordinary Differential Equations''
+Clarendon Press. (1976)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Hamdy 04]{Ham04} Hamdy, S.
+``LiDIA A library for computational number theory''
+Reference manual Edition 2.1.1 May 2004
+\verbwww.cdc.informatik.tudarmstadt.de/TI/LiDIA
\subsection{Proving Axiom Correct} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Adams 99]{Adam99} Adams, A.A.; Gottlieben, H.; Linton, S.A.;
Martin, U.
``Automated theorem proving in support of computer algebra:''
`` symbolic definite integration as a case study''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Adam99.pdf
+\bibitem[Hammarling 85]{Ham85} Hammarling S.
+`` The Singular Value Decomposition in Multivariate Statistics''
+ACM Signum Newsletter. 20, 3 225. (1985)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We assess the current state of research in the application of computer
aided formal reasoning to computer algebra, and argue that embedded
verification support allows users to enjoy its benefits without
wrestling with technicalities. We illustrate this claim by considering
symbolic definite integration, and present a verifiable symbolic
definite integral table look up: a system which matches a query
comprising a definite integral with parameters and side conditions,
against an entry in a verifiable table and uses a call to a library of
lemmas about the reals in the theorem prover PVS to aid in the
transformation of the table entry into an answer. We present the full
model of such a system as well as a description of our prototype
implementation showing the efficacy of such a system: for example, the
prototype is able to obtain correct answers in cases where computer
algebra systems [CAS] do not. We extend upon Fateman's webbased table
by including parametric limits of integration and queries with side
conditions.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Adams 01]{Adam01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne;
Kelsey, Tom; Martin, Ursula; Owre, Sam
``Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS''
\verbwww.csl.sri.com/~owre/papers/tphols01/tphols01.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Adam01.pdf
+\bibitem[Hammersley 67]{HH67} Hammersley J M; Handscomb D C.
+``MonteCarlo Methods''
+Methuen. (1967)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe an interface between version 6 of the Maple computer
algebra system with the PVS automated theorem prover. The interface is
designed to allow Maple users access to the robust and checkable proof
environment of PVS. We also extend this environment by the provision
of a library of proof strategies for use in real analysis. We
demonstrate examples using the interface and the real analysis
library. These examples provide proofs which are both illustrative and
applicable to genuine symbolic computation problems.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{Mahb06,
 author = "Mahboubi, Assia",
 title = "Proving Formally the Implementation of an Efficient gcd Algorithm for Polynomials",
 journal = "Lecture Notes in Computer Science",
 volume = "4130",
 year = "2006",
 pages = "438452",
 paper = "Mahb06.pdf"
+@misc{Hath1896,
+ author = "Hathway, Arthur S.",
+ title = "A Primer Of Quaternions",
+ year = "1896"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe here a formal proof in the Coq system of the structure
theorem for subresultants which allows to prove formally the
correctness of our implementation of the subresultants algorithm.
Up to our knowledge it is the first mechanized proof of this result.
\end{adjustwidth}
+\begin{chunk}{axiom.bib}
+@book{Haya05,
+ author = "Hayashi, K. and Kangkook, J. and Lascu, O. and Pienaar, H. and
+ Schreitmueller, S. and Tarquinio, T. and Thompson, J.",
+ title = "AIX 5L Practical Performance Tools and Tuning Guide",
+ publisher = "IBM",
+ year = "2005",
+ url = "http://www.redbooks.ibm.com/redbooks/pdfs/sg246478.pdf",
+ paper = "Haya05.pdf"
+}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ballarin 99]{Ball99} Ballarin, Clemens; Paulson, Lawrence C.
``A Pragmatic Approach to Extending Provers by Computer Algebra  with Applications to Coding Theory''
\verbwww.cl.cam.ac.uk/~lp15/papers/Isabelle/coding.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Ball99.pdf
+\bibitem[Hayes 70]{Hay70} Hayes J G.
+``Curve Fitting by Polynomials in One Variable''
+Numerical Approximation to Functions and Data.
+(ed J G Hayes) Athlone Press, London. (1970)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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 proofs depend 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 formal specifications for the algorithms, and links the
computer algebra library Sumit to the prover Isabelle.
+\begin{chunk}{ignore}
+\bibitem[Hayes 74]{Hay74} Hayes J G.
+``Numerical Methods for Curve and Surface Fitting''
+Bull Inst Math Appl. 10 144152. (1974)
We give details of the interface, the use of the computer algebra
system on the tacticlevel of Isabelle and its integration into proof
procedures.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bertot 04]{Bert04} Bertot, Yves; Cast\'eran, Pierre
``Interactive Theorem Proving and Program Development''
Springer ISBN 3540208542
+\bibitem[Hayes 74a]{HH74} Hayes J G.; Halliday J,
+``The Leastsquares Fitting of Cubic Spline Surfaces to General Data Sets''
+J. Inst. Math. Appl. 14 89103. (1974)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Coq is an interactive proof assistant for the development of
mathematical theories and formally certified software. It is based on
a theory called the calculus of inductive constructions, a variant of
type theory.
+\begin{chunk}{ignore}
+\bibitem[Henrici 56]{Hen56} Henrici, Peter
+``Automatic Computations with Power Series''
+Journal of the Association for Computing Machinery, Volume 3, No. 1,
+January 1956, 1015
This book provides a pragmatic introduction to the development of
proofs and certified programs using Coq. With its large collection of
examples and exercies it is an invaluable tool for researchers,
students, and engineers interested in formal methods and the
development of zerofault software.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Boulme 00]{BHR00} Boulm\'e, S.; Hardin, T.; Rioboo, R.
``Polymorphic Data Types, Objects, Modules and Functors,: is it too much?''
%\verbaxiomdeveloper.org/axiomwebsite/papers/BHR00.pdf
+\bibitem[Higham 88]{Hig88} Higham, N.J.
+``FORTRAN codes for estimating the onenorm of a
+real or complex matrix, with applications to condition estimation''
+ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381396, December 1988.
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Abstraction is a powerful tool for developers and it is offered by
numerous features such as polymorphism, classes, modules, and
functors, $\ldots$ A working programmer may be confused by this
abundance. We develop a computer algebra library which is being
certificed. Reporting this experience made with a language (Ocaml)
offering all these features, we argue that the are all needed
together. We compare several ways of using classes to represent
algebraic concepts, trying to follow as close as possible mathematical
specification. Thenwe show how to combine classes and modules to
produce code having very strong typing properties. Currently, this
library is made of one hundred units of functional code and behaves
faster than analogous ones such as Axiom.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Boulme 01]{BHHMR01}
Boulm\'e, S.; Hardin, T.; Hirschkoff, D.; M\'enissierMorain, V.; Rioboo, R.
``On the way to certify Computer Algebra Systems''
Calculemus2001
%\verbaxiomdeveloper.org/axiomwebsite/papers/BHHMR01.pdf
+\bibitem[Higham 02]{Hig02} Higham, Nicholas J.
+``Accuracy and stability of numerical algorithms''
+SIAM Philadelphia, PA ISBN 0898715210 (2002)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The FOC project aims at supporting, within a coherent software system,
the entire process of mathematical computation, starting with proved
theories, ending with certified implementations of algorithms. In this
paper, we explain our design requirements for the implementation,
using polynomials as a running example. Indeed, proving correctness of
implementations depends heavily on the way this design allows
mathematical properties to be truly handled at the programming level.
+\begin{chunk}{ignore}
+\bibitem[Hock 81]{HS81} Hock W.; Schittkowski K.
+``Test Examples for Nonlinear Programming Codes''
+Lecture Notes in Economics and Mathematical Systems. 187 SpringerVerlag. 1981
The FOC project, started at the fall of 1997, is aimed to build a
programming environment for the development of certified symbolic
computation. The working languages are Coq and Ocaml. In this paper,
we present first the motivations of the project. We then explain why
and how our concern for proving properties of programs has led us to
certain implementation choices in Ocaml. This way, the sources express
exactly the mathematical dependencies between different structures.
This may ease the achievement of proofs.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Daly 10]{Daly10} Daly, Timothy
``Intel Instruction Semantics Generator''
\verbdaly.axiomdeveloper.org/TimothyDaly_files/publications/sei/intel/intel.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Daly10.pdf
+\bibitem[Householder 70]{Hou70} Householder A S.
+``The Numerical Treatment of a Single Nonlinear Equation''
+McGrawHill. (1970)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Given an Intel x86 binary, extract the semantics of the instruction
stream as Conditional Concurrent Assignments (CCAs). These CCAs
represent the semantics of each individual instruction. They can be
composed to represent higher level semantics.
\end{adjustwidth}
+\begin{chunk}{axiom.bib}
+@book{Hous81,
+ author = "Householder, Alston S.",
+ title = "Principles of Numerical Analysis",
+ publisher = "Dover Publications, Mineola, NY",
+ year = "1981",
+ isbn = "048645312X"
+}
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Danielsson 06]{Dani06} Danielsson, Nils Anders; Hughes, John;
Jansson, Patrik; Gibbons, Jeremy
``Fast and Loose Reasoning is Morally Correct''
ACM POPL'06 January 2005, Charleston, South Carolina, USA
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dani06.pdf
+\bibitem[Huang 96]{HI96} Huang, M.D.; Ierardi, D.
+``Efficient algorithms for RiemannRoch problem and for addition in the
+jacobian of a curve''
+Proceedings 32nd Annual Symposium on Foundations of Computer Sciences.
+IEEE Comput. Soc. Press, pp. 678687.
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Functional programmers often reason about programs as if they were
written in a total language, expecting the results to carry over to
nontoal (partial) languages. We justify such reasoning.
+\subsection{I} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Two languages are defined, one total and one partial, with identical
syntax. The semantics of the partial language includes partial and
infinite values, and all types are lifted, including the function
spaces. A partial equivalence relation (PER) is then defined, the
domain of which is the total subset of the partial language. For types
not containing function spaces the PER relates equal values, and
functions are related if they map related values to related values.
+\begin{chunk}{ignore}
+\bibitem[IBM]{IBM}.
+SCRIPT Mathematical Formula Formatter User's Guide, SH206453,
+IBM Corporation, Publishing Systems Information Development,
+Dept. G68, P.O. Box 1900, Boulder, Colorado, USA 803019191.
It is proved that if two closed terms have the same semantics in the
total language, then they have related semantics in the partial
language. It is also shown that the PER gives rise to a bicartesian
closed category which can be used to reason about values in the domain
of the relation.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 12]{Davenp12} Davenport, James H.; Bradford, Russell;
England, Matthew; Wilson, David
``Program Verification in the presence of complex numbers, functions with
branch cuts etc.''
\verbarxiv.org/pdf/1212.5417.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Davenp12.pdf
+\bibitem[Itoh 88]{Itoh88} Itoh, T.;, Tsujii, S.
+``A fast algorithm for computing multiplicative inverses
+in $GF(2^m)$ using normal bases''
+Inf. and Comp. 78, pp.171177, 1988
+%\verbaxiomdeveloper.org/axiomwebsite/Itoh88.pdf
+ abstract = "
+ This paper proposes a fast algorithm for computing multiplicative
+ inverses in $GF(2^m)$ using normal bases. Normal bases have the
+ following useful property: In the case that an element $x$ in
+ $GF(2^m)$ is represented by normal bases, $2^k$ power operation of an
+ element $x$ in $GF(2^m)$ can be carried out by $k$ times cyclic shift
+ of its vector representation. C.C. Wang et al. proposed an algorithm
+ for computing multiplicative inverses using normal bases, which
+ requires $(m2)$ multiplications in $GF(2^m)$ and $(m1)$ cyclic
+ shifts. The fast algorithm proposed in this paper also uses normal
+ bases, and computes multiplicative inverses iterating multiplications
+ in $GF(2^m)$. It requires at most $2[log_2(m1)]$ multiplications in
+ $GF(2^m)$ and $(m1)$ cyclic shifts, which are much less than those
+ required in Wang's method. The same idea of the proposed fast
+ algorithm is applicable to the general power operation in $GF(2^m)$
+ and the computation of multiplicative inverses in $GF(q^m)$
+ $(q=2^n)$."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In considering the reliability of numerical programs, it is normal to
``limit our study to the semantics dealing with numerical precision''.
On the other hand, there is a great deal of work on the reliability of
programs that essentially ignores the numerics. The thesis of this
paper is that there is a class of problems that fall between these
two, which could be described as ``does the lowlevel arithmeti
implement the highlevel mathematics''. Many of these problems arise
because mathematics, particularly the mathematics of the complex
numbers, is more difficult than expected: for example the complex
function log is not continuous, writing down a program to compute an
inverse function is more complicated than just solving an equation,
and many algebraic simplification rules are not universally valid.
+\begin{chunk}{ignore}
+\bibitem[Iyanaga 77]{Iya77} Iyanaga, Shokichi; Iyanaga, Yukiyosi Kawada
+``Encyclopedic Dictionary of Mathematics''
+1977
+
+\end{chunk}
The good news is that these problems are {\sl theoretically} capable
of being solved, and are {\sl practically} close to being solved, but
not yet solved, in several realworld examples. However, there is
still a long way to go before implementations match the theoretical
possibilities.
\end{adjustwidth}
+\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Dolzmann 97]{Dolz97} Dolzmann, Andreas; Sturm, Thomas
``Guarded Expressions in Practice''
\verbredlog.dolzmann.de/papers/pdf/MIP9702.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dolz97.pdf
+\bibitem[Jacobson 68]{Jac68} Jacobson, N.
+``Structure and Representations of Jordan Algebras''
+AMS, Colloquium Publications Volume 39
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Computer algebra systems typically drop some degenerate cases when
evaluating expressions, e.g. $x/x$ becomes 1 dropping the case
$x=0$. We claim that it is feasible in practice to compute also the
degenerate cases yielding {\sl guarded expressions}. We work over real
closed fields but our ideas about handling guarded expressions can be
easily transferred to other situations. Using formulas as guards
provides a powerful tool for heuristically reducing the combinatorial
explosion of cases: equivalent, redundant, tautological, and
contradictive cases can be detected by simplification and quantifier
elimination. Our approach allows to simplify the expressions on the
basis of simplification knowledge on the logical side. The method
described in this paper is implemented in the REDUCE package GUARDIAN,
which is freely available on the WWW.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[James 81]{JK81} James, Gordon; Kerber, Adalbert
+``The Representation Theory of the Symmetric Group''
+Encyclopedia of Mathematics and its Applications Vol. 16
+AddisonWesley, 1981
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Dos Reis 11]{DR11} Dos Reis, Gabriel; Matthews, David; Li, Yue
``Retargeting OpenAxiom to Poly/ML: Towards an Integrated Proof Assistants
and Computer Algebra System Framework''
Calculemus (2011) Springer
\verbparadise.caltech.edu/~yli/paper/oapolyml.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/DR11.pdf
+\bibitem[Jaswon 77]{JS77} Jaswon, M A.; Symm G T.
+``Integral Equation Methods in Potential Theory and Elastostatics''
+Academic Press. (1977)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper presents an ongoing effort to integrate the Axiom family of
computer algebra systems with Poly/MLbased proof assistants in the
same framework. A long term goal is to make a large set of efficient
implementations of algebraic algorithms available to popular proof
assistants, and also to bring the power of mechanized formal
verification to a family of strongly typed computer algebra systems at
a modest cost. Our approach is based on retargeting the code generator
of the OpenAxiom compiler to the Poly/ML abstract machine.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Dunstan 00a]{Dun00a} Dunstan, Martin N.
``Adding Larch/Aldor Specifications to Aldor''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dunxx.pdf
+\bibitem[Jeffrey 04]{Je04} Jeffrey, Alan
+``Handbook of Mathematical Formulas and Integrals''
+Third Edition, Elsevier Academic Press ISBN 0123822564
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe a proposal to add Larchstyle annotations to the Aldor
programming language, based on our PhD research. The annotations
are intended to be machinecheckable and may be used for a variety
of purposes ranging from compiler optimizations to verification
condition (VC) generation. In this report we highlight the options
available and describe the changes which would need to be made to
the compiler to make use of this technology.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Dunstan 98]{Dun98} Dunstan, Martin; Kelsey, Tom; Linton, Steve;
Martin, Ursula
``Lightweight Formal Methods For Computer Algebra Systems''
\verbwww.cs.standrews.ac.uk/~tom/pub/issac98.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dun98.pdf
+\bibitem[Jenning 66]{Jen66} Jennings A
+``A Compact Storage Scheme for the Solution of Symmetric Linear
+Simultaneous Equations''
+Comput. J. 9 281285. (1966)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Demonstrates the use of formal methods tools to provide a semantics for
the type hierarchy of the Axiom computer algebra system, and a methodology
for Aldor program analysis and verification. There are examples of
abstract specifications of Axiom primitives.
\end{adjustwidth}
+\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Dunstan 99a]{Dun99a} Dunstan, MN
``Larch/Aldor  A Larch BISL for AXIOM and Aldor''
PhD Thesis, 1999
\verbwww.cs.standrews.uk/files/publications/Dun99.php
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dun99a.pdf
+\bibitem[Kalkbrener 91]{Kal91} Kalkbrener, M.
+``Three contributions to elimination theory''
+Ph. D. Thesis, University of Linz, Austria, 1991
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this thesis we investigate the use of lightweight formal methods
and verification conditions (VCs) to help improve the reliability of
components constructed within a computer algebra system. We follow the
Larch approach to formal methods and have designed a new behavioural
interface specification language (BISL) for use with Aldor: the
compiled extension language of Axiom and a fullyfeatured programming
language in its own right. We describe our idea of lightweight formal
methods, present a design for a lightweight verification condition
generator and review our implementation of a prototype verification
condition generator for Larch/Aldor.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Dunstan 00]{Dun00} Dunstan, Martin; Kelsey, Tom; Martin, Ursula;
Linton, Steve
``Formal Methods for Extensions to CAS''
FME'99, Toulouse, France, Sept 2024, 1999, pp 17581777
\verbtom.host.cs.standrews.ac.uk/pub/fm99.ps
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dun00.pdf
+\bibitem[Kalkbrener 98]{Kal98} Kalkbrener, M.
+``Algorithmic properties of polynomial rings''
+Journal of Symbolic Computation 1998
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We demonstrate the use of formal methods tools to provide a semantics
for the type hierarchy of the AXIOM computer algebra system, and a
methodology for Aldor program analysis and verification. We give a
case study of abstract specifications of AXIOM primitives, and provide
an interface between these abstractions and Aldor code.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@misc{Hard13,
 author = "Hardin, David S. and McClurg, Jedidiah R. and Davis, Jennifer A.",
 title = "Creating Formally Verified Components for Layered Assurance with an LLVM to ACL2 Translator",
 url = "http://www.jrmcclurg.com/papers/law_2013_paper.pdf",
 paper = "Hard13.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Kantor 89]{Kan89} Kantor,I.L.; Solodovnikov, A.S.
+``Hypercomplex Numbers''
+Springer Verlag Heidelberg, 1989, ISBN 0387969802
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper describes an effort to create a library of formally
verified software component models from code that have been compiled
using the LowLevel Virtual Machine (LLVM) intermediate form. The idea
is to build a translator from LLVM to the applicative subset of Common
Lisp accepted by the ACL2 theorem prover. They perform verification of
the component model using ACL2's automated reasoning capabilities.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@misc{Hard14,
 author = "Hardin, David S. and Davis, Jennifer A. and Greve, David A. and McClurg, Jedidiah R.",
 title = "Development of a Translator from LLVM to ACL2",
 url = "http://arxiv.org/pdf/1406.1566",
 paper = "Hard14.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Kaufmann 00]{KMJ00} Kaufmann, Matt; Manolios, Panagiotis;
+Moore J Strother
+``ComputerAided Reasoning: An Approach''
+Springer, July 31. 2000 ISBN 0792377443
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In our current work a library of formally verified software components
is to be created, and assembled, using the LowLevel Virtual Machine
(LLVM) intermediate form, into subsystems whose toplevel assurance
relies on the assurance of the individual components. We have thus
undertaken a project to build a translator from LLVM to the
applicative subset of Common Lisp accepted by the ACL2 theorem
prover. Our translator produces executable ACL2 formal models,
allowing us to both prove theorems about the translated models as well
as validate those models by testing. The resulting models can be
translated and certified without user intervention, even for code with
loops, thanks to the use of the def::ung macro which allows us to
defer the question of termination. Initial measurements of concrete
execution for translated LLVM functions indicate that performance is
nearly 2.4 million LLVM instructions per second on a typical laptop
computer. In this paper we overview the translation process and
illustrate the translator's capabilities by way of a concrete example,
including both a functional correctness theorem as well as a
validation test for that example.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Lamport 02]{Lamp02} Lamport, Leslie
``Specifying Systems''
\verbresearch.microsoft.com/enus/um/people/lamport/tla/book020808.pdf
AddisonWesley ISBN 032114306X
%\verbaxiomdeveloper.org/axiomwebsite/papers/Lamp02.pdf
+\bibitem[Knuth 71]{Knu71} Knuth, Donald
+``The Art of Computer Programming''
+2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing,
+AddisonWesley 1971, p. 397398
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Martin 97]{Mart97} Martin, U.; Shand, D.
``Investigating some Embedded Verification Techniques for Computer Algebra Systems''
\verbwww.risc.jku.at/conferences/Theorema/papers/shand.ps.gz
%\verbaxiomdeveloper.org/axiomwebsite/papers/Mart97.ps
+\bibitem[Knuth 84]{Knu84} Knuth, Donald
+{\it The \TeX{}book}.
+Reading, Massachusetts, AddisonWesley Publishing Company, Inc.,
+1984. ISBN 0201134489
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
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{adjustwidth}

\begin{chunk}{axiom.bib}
@book{Maso86,
 author = "Mason, Ian A.",
 title = "The Semantics of Destructive Lisp",
 publisher = "Center for the Study of Language and Information",
 year = "1986",
 isbn = "0937073067"
}
+@book{Knut92,
+ author = "Knuth, Donald E.",
+ title = "Literate Programming",
+ publisher = "Center for the Study of Language and Information, Stanford CA",
+ year = "1992",
+ isbn = "0937073814"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Our basic premise is that the ability to construct and modify programs
will not improve without a new and comprehensive look at the entire
programming process. Past theoretical research, say, in the logic of
programs, has tended to focus on methods for reasoning about
individual programs; little has been done, it seems to us, to develop
a sound understanding of the process of programming  the process by
which programs evolve in concept and in practice. At present, we lack
the means to describe the techniques of program construction and
improvement in ways that properly link verification, documentation and
adaptability.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Newcombe 13]{Newc13} Newcombe, Chris; Rath, Tim; Zhang, Fan;
Munteanu, Bogdan; Brooker, Marc; Deardeuff, Michael
``Use of Formal Methods at Amazon Web Services''
\verbresearch.microsoft.com/enus/um/people/lamport/tla/
\verbformalmethodsamazon.pdf
+\bibitem[Knu98]{Knu98} Donald Knuth
+``The Art of Computer Programming'' Vol. 3
+(Sorting and Searching)
+AddisonWesley 1998
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In order to find subtle bugs in a system design, it is necessary to
have a precise description of that design. There are at least two
major benefits to writing a precise design; the author is forced to
think more clearly, which helps eliminate ``plausible handwaving'',
and tools can be applied to check for errors in the design, even while
it is being written. In contrast, conventional design documents
consist of prose, static diagrams, and perhaps pseudocode in an ad
hoc untestable language. Such descriptions are far from precise; they
are often ambiguous, or omit critical aspects such as partial failure
or the granularity of concurrency (i.e. which constructs are assumed
to be atomic). At the other end of the spectrum, the final executable
code is unambiguous, but contains an overwhelming amount of detail. We
needed to be able to capture the essence of a design in a few hundred
lines of precise description. As our designs are unavoidably complex,
we need a highlyexpressive language, far above the level of code, but
with precise semantics. That expressivity must cover realworld
concurrency and faulttolerance. And, as we wish to build services
quickly, we wanted a language that is simple to learn and apply,
avoiding esoteric concepts. We also very much wanted an existing
ecosystem of tools. We found what we were looking for in TLA+, a
formal specification language.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Poll 99a]{P99a} Poll, Erik
``The Type System of Axiom''
\verbwww.cs.ru.nl/E.Poll/talks/axiom.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/P99a.pdf
+\bibitem[Kobayashi 89]{Koba89} Kobayashi, H.; Moritsugu, S.; Hogan, R.W.
+``On Radical ZeroDimensional Ideals''
+J. Symbolic Computations 8, 545552 (1989)
+\verbwww.sciencedirect.com/science/article/pii/S0747717189800604/pdf
+\verb?md5=f06dc6269514c90dcae57f0184bcbe65&
+\verbpid=1s2.0S0747717189800604main.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Koba88.pdf
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This is a slide deck from a talk on the correspondence between
Axiom/Aldor types and Logic.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Poll 99]{PT99} Poll, Erik; Thompson, Simon
``The Type System of Aldor''
\verbwww.cs.kent.ac.uk/pubs/1999/874/content.ps
%\verbaxiomdeveloper.org/axiomwebsite/papers/PT99.pdf
+\bibitem[Kolchin 73]{Kol73} Kolchin, E.R.
+``Differential Algebra and Algebraic Groups''
+(Academic Press, 1973).
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper gives a formal description of  at least a part of 
the type system of Aldor, the extension language of the Axiom.
In the process of doing this a critique of the design of the system
emerges.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Poll (a)]{PTxx} Poll, Erik; Thompson, Simon
``Adding the axioms to Axiom. Toward a system of automated reasoning in
Aldor''
\verbciteseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.1457&rep=rep1&type=ps
%\verbaxiomdeveloper.org/axiomwebsite/papers/PTxx.pdf
+\bibitem[Koutschan 10]{Kou10} Koutschan, Christoph
+``Axiom / FriCAS''
+\verbwww.risc.jku.at/education/courses/ws2010/cas/axiom.pdf
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper examines the proposal of using the type system of Axiom to
represent a logic, and thus to use the constructions of Axiom to
handle the logic and represent proofs and propositions, in the same
way as is done in theorem provers based on type theory such as Nuprl
or Coq.

The paper shows an interesting way to decorate Axiom with pre and
postconditions.

The CurryHoward correspondence used is
\begin{verbatim}
PROGRAMMING LOGIC
Type Formula
Program Proof
Product/record type (...,...) Conjunction
Sum/union type \/ Disjunction
Function type > Implication
Dependent function type (x:A) > B(x) Universal quantifier
Dependent product type (x:A,B(x)) Existential quantifier
Empty type Exit Contradictory proposition
One element type Triv True proposition
\end{verbatim}
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Poll 00]{PT00} Poll, Erik; Thompson, Simon
``Integrating Computer Algebra and Reasoning through the Type System
of Aldor''
%\verbaxiomdeveloper.org/axiomwebsite/papers/PT00.pdf
+\bibitem[Kozen 86]{KL86} Kozen, Dexter; Landau, Susan
+``Polynomial Decomposition Algorithms''
+Journal of Symbolic Computation (1989) 7, 445456
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A number of combinations of reasoning and computer algebra systems
have been proposed; in this paper we describe another, namely a way to
incorporate a logic in the computer algebra system Axiom. We examine
the type system of Aldor  the Axiom Library Compiler  and show
that with some modifications we can use the dependent types of the
system to model a logic, under the CurryHoweard isomorphism. We give
a number of example applications of the logi we construct and explain
a prototype implementation of a modified typechecking system written
in Haskell.
\end{adjustwidth}
+\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interval Arithmetic} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Boehm 86]{Boe86} Boehm, HansJ.; Cartwright, Robert; Riggle, Mark;
O'Donnell, Michael J.
``Exact Real Arithmetic: A Case Study in Higher Order Programming''
\verbdev.acm.org/pubs/citations/proceedings/lfp/319838/p162boehm
%\verbaxiomdeveloper.org/axiomwebsite/papers/Boe86.pdf
+\begin{chunk}{axiom.bib}
+@book{Lamp86,
+ author = "Lamport, Leslie",
+ title = "LaTeX: A Document Preparation System",
+ publisher = "AddisonWesley Publishing Company, Reading, Massachusetts",
+ year = "1986",
+ isbn = "020115790X"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Briggs 04]{Bri04} Briggs, Keith
``Exact real arithmetic''
\verbkeithbriggs.info/documents/xrkenttalkpp.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bri04.pdf
+\bibitem[Lautrup 71]{Lau71} Lautrup B.
+``An Adaptive Multidimensional Integration Procedure''
+Proc. 2nd Coll. on Advanced Methods in Theoretical Physics, Marseille. (1971)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fateman 94]{Fat94} Fateman, Richard J.; Yan, Tak W.
``Computation with the Extended Rational Numbers and an Application to
Interval Arithmetic''
\verbwww.cs.berkeley.edu/~fateman/papers/extrat.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat94.pdf
+\bibitem[Lawson 77]{Law77} Lawson C L.
+``Software for C Surface Interpolation''
+Mathematical Software III. (ed J R Rice) Academic Press. 161194. (1977)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Programming languages such as Common Lisp, and virtually every
computer algebra system (CAS), support exact arbitraryprecision
integer arithmetic as well as exect rational number computation.
Several CAS include interval arithmetic directly, but not in the
extended form indicated here. We explain why changes to the usual
rational number system to include infinity and ``notanumber'' may be
useful, especially to support robust interval computation. We describe
techniques for implementing these changes.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Lawson 74]{LH74} Lawson C L.; Hanson R J.
+``Solving Leastsquares Problems''
+PrenticeHall. (1974)
+
+\end{chunk}
\begin{chunk}{axiom.bib}
@incollection{Lamb06,
 author = "Lambov, Branimir",
 title = "Interval Arithmetic Using SSE2",
 booktitle = "Lecture Notes in Computer Science",
 publisher = "SpringerVerlag",
 year = "2006",
 isbn = "9783540855200",
 pages = "102113"
+@article{Laws79,
+ author = "Lawson, C.L. and Hanson R.J. and Kincaid, D.R. and Krogh, F.T.",
+ title = "Algorithm 539: Basic linear algebra subprograms for FORTRAN usage",
+ journal = "ACM Transactions on Mathematical Software",
+ volume = "5",
+ number = "3",
+ month = "September",
+ year = "1979",
+ pages = "308323"
}
\end{chunk}
\subsection{Numerics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Atkinson 09]{Atk09} Atkinson, Kendall; Han, Welmin; Stewear, David
``Numerical Solution of Ordinary Differential Equations''
\verbhomepage.math.uiowa.edu/~atkinson/papers/NAODE_Book.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Atk09.pdf
+\bibitem[Lawson 79]{LHKK79} Lawson C L; Hanson R J; Kincaid D R;
+ Krogh F T
+``Basic Linear Algebra Subprograms for Fortran Usage''
+ACM Trans. Math. Softw. 5 308325. (1979)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This book is an expanded version of supplementary notes that we used
for a course on ordinary differential equations for upperdivision
undergraduate students and beginning graduate students in mathematics,
engineering, and sciences. The book introduces the numerical analysis
of differential equations, describing the mathematical background for
understanding numerical methods and giving information on what to
expect when using them. As a reason for studying numerical methods as
a part of a more general course on differential equations, many of the
basic ideas of the numerical analysis of differential equations are
tied closely to theoretical behavior associated with the problem being
solved. For example, the criteria for the stability of a numerical
method is closely connected to the stability of the differential
equation problem being solved.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Crank 96]{Cran96} Crank, J.; Nicolson, P.
``A practical method for numerical evaluations of solutions of partial differential equations of heatconduction type''
Advances in Computational Mathematics Vol 6 pp207226 (1996)
\verbwww.acms.arizona.edu/FemtoTheory/MK_personal/opti547/literature/
\verbCNMethodoriginal.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Cran96.pdf
+\bibitem[Lazard 91]{Laz91} Lazard, D.
+``A new method for solving algebraic systems of positive dimension''
+Discr. App. Math. 33:147160,1991
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lef\'evre 06]{Lef06} Lef\'evre, Vincent; Stehl\'e, Damien;
Zimmermann, Paul
``Worst Cases for the Exponential Function
in the IEEE754r decimal64 Format''
in Lecture Notes in Computer Science, Springer ISBN 9783540855200
(2006) pp114125
+\bibitem[Lazard92]{Laz92} Lazard, D.
+``Solving Zerodimensional Algebraic Systems''
+Journal of Symbolic Computation, 1992, 13, 117131
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Laza90,
+ author = "Lazard, Daniel and Rioboo, Renaud",
+ title = "Integration of rational functions: Rational computation of the
+ logarithmic part",
+ journal = "Journal of Symbolic Computation",
+ volume = "9",
+ number = "2",
+ year = "1990",
+ month = "February",
+ pages = "113115",
+ keywords = "axiomref",
+ paper = "Laza90.pdf",
+ abstract = "
+ A new formula is given for the logarithmic part of the integral of a
+ rational function, one that strongly improves previous algorithms and
+ does not need any computation in an algebraic extension of the field
+ of constants, nor any factorisation since only polynomial arithmetic
+ and GCD computations are used. This formula was independently found
+ and implemented in SCRATCHPAD by B.M. Trager."
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We searched for the worst cases for correct rounding of the
exponential function in the IEEE 754r decimal64 format, and computed
all the bad cases whose distance from a breakpoint (for all rounding
modes) is less than $10^{15}$ ulp, and we give the worst ones. In
particular, the worst case for
$\vert{}x\vert{} \ge 3 x 10^{11}$ is
\[
exp(9.407822313572878x10^{2} = 1.09864568206633850000000000000000278\ldots
\]
This work can be extended to other elementary functions in the decimal64
format and allows the design of reasonably fast routines that will
evaluate these functions with correct rounding, at least in some situations.
\end{adjustwidth}
+\begin{chunk}{axiom.bib}
+@article{LeBr88,
+ author = "Le Brigand, D.; Risler, J.J.",
+ title = "Algorithme de BrillNoether et codes de Goppa",
+ journal = "Bull. Soc. Math. France",
+ volume = "116",
+ year = "1988",
+ pages = "231253"
+}
+
+\end{chunk}
\begin{chunk}{axiom.bib}
@book{Hamm62,
 author = "Hamming R W.",
 title = "Numerical Methods for Scientists and Engineers",
 publisher = "Dover",
 year = "1973",
 isbn = "0486652416"
+@book{Lege11,
+ author = "Legendre, George L. and Grazini, Stefano",
+ title = "Pasta by Design",
+ publisher = "Thames and Hudson",
+ isbn = "9780500515808",
+ year = "2011"
}
\end{chunk}
\subsection{Advanced Documentation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem [Bostock 14]{Bos14} Bostock, Mike
``Visualizing Algorithms''
\verbbost.ocks.org/mike/algorithms
+\bibitem[Lenstra 87]{LS87} Lenstra, H. W.; Schoof, R. J.
+``Primitivive Normal Bases for Finite Fields''
+Math. Comp. 48, 1987, pp. 217231
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This website hosts various ways of visualizing algorithms. The hope is
that these kind of techniques can be applied to Axiom.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Leeuwen]{Leexx} van Leeuwen, Andr\'e M.A.
``Representation of mathematical object in interactive books''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Leexx.pdf
+\begin{chunk}{axiom.bib}
+@misc{Leop03,
+ author = "Leopardi, Paul",
+ title = "A quick introduction to Clifford Algebras",
+ publisher = "School of Mathematics, University of New South Wales",
+ year = "2003",
+ paper = "Leop03.pdf"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present a model for the representation of mathematical objects in
structured electronic documents, in a way that allows for interaction
with applications such as computer algebra systems and proof checkers.
Using a representation that reflects only the intrinsic information of
an object, and storing applicationdependent information in socalled
{\sl application descriptions}, it is shown how the translation from
the internal to an external representation and {\sl vice versa} can be
achieved. Hereby a formalisation of the concept of {\sl context} is
introduced. The proposed scheme allows for a high degree of
application integration, e.g., parallel evaluation of subexpressions
(by different computer algebra systems), or a proof checker using a
computer algebra system to verify an equation involving a symbolic
computation.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Soiffer 91]{Soif91} Soiffer, Neil Morrell
``The Design of a User Interface for Computer Algebra Systems''
\verbwww.eecs.berkeley.edu/Pubs/TechRpts/1991/CSD91626.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Soif91.pdf
+\bibitem[Lewis 77]{Lew77} Lewis J G,
+``Algorithms for sparse matrix eigenvalue problems''
+Technical Report STANCS77595. Computer Science Department,
+Stanford University. (1977)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This thesis discusses the design and implementation of natural user
interfaces for Computer Algebra Systems. Such an interface must not
only display expressions generated by the Computer Algebra System in
standard mathematical notation, but must also allow easy manipulation
and entry of expressions in that notation. The user interface should
also assist in understanding of large expressions that are generated
by Computer Algebra Systems and should be able to accommodate new
notational forms.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Victor 11]{Vict11} Victor, Bret
``Up and Down the Ladder of Abstraction''
\verbworrydream.com/LadderOfAbstraction
+\bibitem[Lidl 83]{LN83} Lidl, R.; Niederreiter, H.
+``Finite Field, Encycoldia of Mathematics and Its Applications''
+Vol. 20, Cambridge Univ. Press, 1983 ISBN 0521302404
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This interactive essay presents the ladder of abstraction, a technique for
thinking explicitly about these levels, so a designer can move among
them consciously and confidently.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Victor 12]{Vict12} Victor, Bret
``Inventing on Principle''
\verbwww.youtube.com/watch?v=PUv66718DII
+\bibitem[Linger 79]{LMW79} Linger, Richard C.; Mills, Harlan D.;
+Witt, Bernard I.
+``Structured Programming: Theory and Practice''
+AddisonWesley (March 1979) ISBN 0201144611
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This video raises the level of discussion about humancomputer interaction
from a technical question to a question of effectively capturing ideas.
In particular, this applies well to Axiom's focus on literate programming.
\end{adjustwidth}

\subsection{Differential Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Abramov 95]{Abra95} Abramov, Sergei A.; Bronstein, Manuel;
Petkovsek, Marko
``On Polynomial Solutions of Linear Operator Equations''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Abra95.pdf
+\bibitem[Lipson 81]{Lip81} Lipson, D.
+``Elements of Algebra and Algebraic Computing''
+The Benjamin/Cummings Publishing Company, Inc.Menlo Park, California, 1981.
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Abramov 01]{Abra01} Abramov, Sergei; Bronstein, Manuel
``On Solutions of Linear Functional Systems''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Abra01.pdf
+\begin{chunk}{axiom.bib}
+@misc{Loet09,
+ author = "Loetzsch, Martin and Bleys, Joris and Wellens, Pieter",
+ title = "Understanding the Dynamics of Complex Lisp Programs",
+ year = "2009",
+ url = "http://www.martinloetzsch.de/papers/loetzsch09understanding.pdf",
+ paper = "Loet09.pdf"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe a new direct algorithm for transforming a linear system of
recurrences into an equivalent one with nonsingular leading or
trailing matrix. Our algorithm, which is an improvement to the EG
elimination method, uses only elementary linear algebra operations
(ranks, kernels, and determinants) to produce an equation satisfied by
the degress of the solutions with finite support. As a consequence, we
can boudn and compute the polynomial and rational solutions of very
general linear functional systems such as systems of differential or
($q$)difference equations.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 96b]{Bro96b} Bronstein, Manuel
``On the Factorization of Linear Ordinary Differential Operators''
Mathematics and Computers in Simulation 42 pp 387389 (1996)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro96b.pdf
+\begin{chunk}{axiom.bib}
+@misc{Loet00,
+ author = "Loetzsch, M.",
+ title = "GTFL  A graphical terminal for Lisp",
+ year = "2000",
+ url = "http://martinloetzsch.de/gtfl"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
After reviewing the arithmetic of linear ordinary differential
operators, we describe the current status of the factorisation
algorithm, specially with respect to factoring over nonalgebraically
closed constant fields. We also describe recent results from Singer
and Ulmer that reduce determining the differential Galois group of an
operator to factoring.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 96a]{Bro96a} Bronstein, Manuel; Petkovsek, Marko
``An introduction to pseudolinear algebra''
Theoretical Computer Science V157 pp333 (1966)
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro96a.pdf
+\begin{chunk}{axiom.bib}
+@book{Losc60,
+ author = {L\"osch, Friedrich},
+ title = "Tables of Higher Functions",
+ publisher = "McGrawHill Book Company",
+ year = "1960"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Pseudolinear algebra is the study of common properties of linear
differential and difference operators. We introduce in this paper its
basic objects (pseudoderivations, skew polynomials, and pseudolinear
operators) and describe several recent algorithms on them, which, when
applied in the differential and difference cases, yield algorithms for
uncoupling and solving systems of linear differential and difference
equations in closed form.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein xb]{Broxb} Bronstein, Manuel
``Computer Algebra Algorithms for Linear Ordinary Differential and
Difference equations''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/ecm3.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Broxb.pdf
+\bibitem[LTU10]{LTU10}.
+``Lambda the Ultimate''
+\verblambdatheultimate.org/node/3663#comment62440
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Galois theory has now produced algorithms for solving linear ordinary
differential and difference equations in closed form. In addition,
recent algorithmic advances have made those algorithms effective and
implementable in computer algebra systems. After introducing the
relevant parts of the theory, we describe the latest algorithms for
solving such equations.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 94]{Bro94} Bronstein, Manuel
``An improved algorithm for factoring linear ordinary differential
operators''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
+\begin{chunk}{axiom.bib}
+@book{Luke69a,
+ author = "Luke, Yudell L.",
+ title = "The Special Functions and their Approximations",
+ volume = "1",
+ publisher = "Academic Press",
+ year = "1969",
+ booktitle = "Mathematics in Science and Engineering Volume 53I"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe an efficient algorithm for computing the associated
equations appearing in the BekeSchlesinger factorisation method for
linear ordinary differential operators. This algorithm, which is based
on elementary operations with sets of integers, can be easily
implemented for operators of any order, produces several possible
associated equations, of which only the simplest can be selected for
solving, and often avoids the degenerate case, where the order of the
associated equation is less than in the generic case. We conclude with
some fast heuristics that can produce some factorizations while using
only linear computations.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 90]{Bro90} Bronstein, Manuel
``On Solutions of Linear Ordinary Differential Equations in their
Coefficient Field''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro90.pdf
+\begin{chunk}{axiom.bib}
+@book{Luke69b,
+ author = "Luke, Yudell L.",
+ title = "The Special Functions and their Approximations",
+ volume = "2",
+ publisher = "Academic Press",
+ year = "1969",
+ booktitle = "Mathematics in Science and Engineering Volume 53I"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe a rational algorithm for finding the denominator of any
solution of a linear ordinary differential equation in its coefficient
field. As a consequence, there is now a rational algorithm for finding
all such solutions when the coefficients can be built up from the
rational functions by finitely many algebraic and primitive
adjunctions. This also eliminates one of the computational bottlenecks
in algorithms that either factor or search for Liouvillian solutions
of such equations with Liouvillian coefficients.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 96]{Bro96} Bronstein, Manuel
``$\sum^{IT}$  A stronglytyped embeddable computer algebra library''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro96.pdf
+\bibitem[Lyness 83]{Lyn83} Lyness J N.
+``When not to use an automatic quadrature routine''
+SIAM Review. 25 6387. (1983)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe the new computer algebra library $\sum^{IT}$ and its
underlying design. The development of $\sum^{IT}$ is motivated by the
need to provide highly efficient implementations of key algorithms for
linear ordinary differential and ($q$)difference equations to
scientific programmers and to computer algebra users, regardless of
the programming language or interactive system they use. As such,
$\sum^{IT}$ is not a computer algebra system per se, but a library (or
substrate) which is designed to be ``plugged'' with minimal efforts
into different types of client applications.
\end{adjustwidth}
+\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Bronstein 99a]{Bro99a} Bronstein, Manuel
``Solving linear ordinary differential equations over
$C(x,e^{\int{f(x)dx}})$
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro99a.pdf
+\bibitem[Mac Lane 79]{MB79} Mac Lane, Saunders; Birkhoff, Garret
+``Algebra''
+AMS Chelsea Publishing ISBN 0821816462
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe a new algorithm for computing the solutions in
\[F=C(x,e^{\int{f(x)dx}})\] of linear ordinary differential equations
with coefficients in $F$. Compared to the general algorithm, our
algorithm avoids the computation of exponential solutions of equations
with coefficients in $C(x)$, as well as the solving of linear
differential systems over $C(x)$. Our method is effective and has been
implemented.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 00]{Bro00} Bronstein, Manuel
``On Solutions of Linear Ordinary Differential Equations in their Coefficient Field''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro00.pdf
+\bibitem[Malcolm 72]{Mal72} Malcolm M. A.
+``Algorithms to reveal properties of floatingpoint arithmetic''
+Comms. of the ACM, 15, 949951. (1972)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We extend the notion of monomial extensions of differential fields,
i.e. simple transcendental extensions in which the polynomials are
closed under differentiation, to difference fields. The structure of
such extensions provides an algebraic framework for solving
generalized linear difference equations with coefficients in such
fields. We then describe algorithms for finding the denominator of any
solution of those equations in an important subclass of monomial
extensions that includes transcendental indefinite sums and
products. This reduces the general problem of finding the solutions of
such equations in their coefficient fields to bounding their
degrees. In the base case, this yields in particular a new algorithm
for computing the rational solutions of $q$difference equations with
polynomial coefficients.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 02]{Bro02} Bronstein, Manuel; Lafaille, S\'ebastien
``Solutions of linear ordinary differential equations in terms of
special functions''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro02.pdf
+\bibitem[Malcolm 76]{MS76} Malcolm M A.; Simpson R B.
+``Local Versus Global Strategies for Adaptive Quadrature''
+ACM Trans. Math. Softw. 1 129146. (1976)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We describe a new algorithm for computing special function solutions
of the form $y(x) = m(x)F(\eta(x))$ of second order linear ordinary
differential equations, where $m(x)$ is an arbitrary Liouvillian
function, $\eta(x)$ is an arbitrary rational function, and $F$
satisfies a given second order linear ordinary differential
equations. Our algorithm, which is base on finding an appropriate
point transformation between the equation defining $F$ and the one to
solve, is able to find all rational transformations for a large class
of functions $F$, in particular (but not only) the $_0F_1$ and $_1F_1$
special functions of mathematical physics, such as Airy, Bessel,
Kummer and Whittaker functions. It is also able to identify the values
of the parameters entering those special functions, and can be
generalized to equations of higher order.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 03]{Bro03} Bronstein, Manuel; Trager, Barry M.
``A Reduction for Regular Differential Systems''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mega2003.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro03.pdf
+\bibitem[Marden 66]{Mar66} Marden M.
+``Geometry of Polynomials''
+Mathematical Surveys. 3 Am. Math. Soc., Providence, RI. (1966)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We propose a definition of regularity of a linear differential system
with coefficients in a monomial extension of a differential field, as
well as a global and truly rational (i.e. factorisationfree)
iteration that transforms a system with regular finite singularites
into an equivalent one with simple finite poles. We then apply our
iteration to systems satisfied by bases of algebraic function fields,
obtaining algorithms for computing the number of irreducible
components and the genus of algebraic curves.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 03a]{Bro03a} Bronstein, Manuel; Sol\'e, Patrick
``Linear recurrences with polynomial coefficients''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro03a.pdf
+\begin{chunk}{axiom.bib}
+@misc{Mars07,
+ author = "Marshak, U.",
+ title = "HTAJAX  AJAX framework for Hunchentoot",
+ year = "2007",
+ url = "http://commonlisp.net/project/htajax/htajax.html"
+}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We relate sequences generated by recurrences with polynomial
coefficients to interleaving and multiplexing of sequences generated
by recurrences with constant coefficients. In the special case of
finite fields, we show that such sequences are periodic and provide
linear complexity estimates for all three constructions.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 05]{Bro05} Bronstein, Manuel; Li, Ziming; Wu, Min
``PicardVessiot Extensions for Linear Functional Systems''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/publications/issac2005.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro05.pdf
+\bibitem[Maza 95]{MR95} Maza, M. Moreno; Rioboo, R.
+``Computations of gcd over algebraic towers of simple extensions''
+In proceedings of AAECC11 Paris, 1995.
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
PicardVessiot extensions for ordinary differential and difference
equations are well known and are at the core of the associated Galois
theories. In this paper, we construct fundamental matrices and
PicardVessiot extensions for systems of linear partial functional
equations having finite linear dimension. We then use those extensions
to show that all the solutions of a factor of such a system can be
completed to solutions of the original system.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Davenport 86]{Dav86} Davenport, J.H.
``The Risch Differential Equation Problem''
SIAM J. COMPUT. Vol 15, No. 4 1986
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav86.pdf
+\bibitem[Maza 97]{Maz97} Maza, M. Moreno
+``Calculs de pgcd audessus des tours
+d'extensions simples et resolution des systemes d'equations algebriques''
+These, Universite P.etM. Curie, Paris, 1997.
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We propose a new algorithm, similar to Hermite's method for the
integration of rational functions, for the resolution of Risch
differential equations in closed form, or proving that they have no
resolution. By requiring more of the presentation of our differential
fields (in particular that the exponentials be weakly normalized), we
can avoid the introduction of arbitrary constants which have to be
solved for later.
+\begin{chunk}{ignore}
+\bibitem[Maza 98]{Maz98} Maza, M. Moreno
+``A new algorithm for computing triangular
+decomposition of algebraic varieties''
+ NAG Tech. Rep. 4/98.
We also define a class of fields known as exponentially reduced, and
show that solutions of Risch differential equations which arise from
integrating in these fields satisfy the ``natural'' degree constraints
in their main variables, and we conjecture (after Risch and Norman)
that this is true in all variables.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Singer 9]{Sing91.pdf} singer, Michael F.
``Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients''
J. Symbolic Computation V11 No 3 pp251273 (1991)
\verbwww.sciencedirect.com/science/article/pii/S074771710880048X
%\verbaxiomdeveloper.org/axiomwebsite/papers/Sing91.pdf
+\bibitem[Mignotte 82]{Mig82} Mignotte, Maurice
+``Some Useful Bounds''
+Computing, Suppl. 4, 259263 (1982), SpringerVerlag
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Let $L(y)=b$ be a linear differential equation with coefficients in a
differential field $K$. We discuss the problem of deciding if such an
equation has a nonzero solution in $K$ and give a decision procedure
in case $K$ is an elementary extension of the field of rational
functions or is an algebraic extension of a transcendental liouvillian
extension of the field of rational functions We show how one can use
this result to give a procedure to find a basis for the space of
solutions, liouvillian over $K$, of $L(y)=0$ where $K$ is such a field
and $L(y)$ has coefficients in $K$.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Von Mohrenschildt 94]{Mohr94} Von Mohrenschildt, Martin
``Symbolic Solutions of Discontinuous Differential Equations''
\verbecollection.library.ethz.ch/eserv/eth:39463/eth3946301.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Mohr94.pdf
+\bibitem[McCarthy 83]{McC83} McCarthy G J.
+``Investigation into the Multigrid Code MGD1''
+Report AERER 10889. Harwell. (1983)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Von Mohrenschildt 98]{Mohr98} Von Mohrenschildt, Martin
``A Normal Form for Function Rings of Piecewise Functions''
J. Symbolic Computation (1998) Vol 26 pp607619
\verbwww.cas.mcmaster.ca/~mohrens/JSC.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Mohr98.pdf
+\bibitem[Mie97]{Mie97} Mielenz, Klaus D.
+``Computation of Fresnel Integrals''
+J. Res. Natl. Inst. Stand. Technol. (NIST) V102 No3 MayJune 1997 pp363365
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Computer algebra systems often have to deal with piecewise continuous
functions. These are, for example, the absolute value function,
signum, piecewise defined functions but also functions that are the
supremum or infimum of two functions. We present a new algebraic
approach to these types of problems. This paper presents a normal form
for a function ring containing piecewise polynomial functions of an
expression. The main result is that this normal form can be used to
decide extensional equality of two piecewise functions. Also we define
supremum and infimum for piecewise functions; in fact, we show that
the function ring forms a lattice. Additionally, a method to solve
equalities and inequalities in this function ring is
presented. Finally, we give a ``user interface'' to the algebraic
representation of the piecewise functions.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Weber 06]{Webe06} Weber, Andreas
``Quantifier Elimination on Real Closed Fields and Differential Equations''
\verbcg.cs.unibonn.de/personalpages/weber/publications/pdf/WeberA/Weber2006a.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Webe06.pdf
 keywords = "survey",
+\bibitem[Mie00]{Mie00} Mielenz, Klaus D.
+``Computation of Fresnel Integrals II''
+J. Res. Natl. Inst. Stand. Technol. (NIST) V105 No4 JulyAug 2000 pp589590
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper surveys some recent applications of quantifier elimination
on real closed fields in the context of differential
equations. Although polynomial vector fields give rise to solutions
involving the exponential and other transcendental functions in
general, many questions can be settled within the real closed field
without referring to the real exponential field. The technique of
quantifier elimination on real closed fields is not only of
theoretical interest, but due to recent advances on the algorithmic
side including algorithms for the simplification of quantifierfree
formulae the method has gained practical applications, e.g. in the
context of computing threshold conditions in epidemic modeling.
\end{adjustwidth}{2.5em}{0pt}

\begin{chunk}{ignore}
\bibitem[Ulmer 03]{Ulm03} Ulmer, Felix
``Liouvillian solutions of third order differential equations''
J. Symbolic COmputations 36 pp 855889 (2003)
\verbwww.sciencedirect.com/science/article/pii/S0747717103000658
%\verbaxiomdeveloper.org/axiomwebsite/papers/Ulm03.pdf
+\bibitem[Millen 68]{Mil68} Millen, J. K.
+``CHARYBDIS: A LISP program to display mathematical expressions on
+typewriterlike devices''
+Interactive Systems for Experimental and Applied Mathematics
+M. Klerer and J. Reinfelds, eds., Academic Press, New York 1968, pp7990
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Mil68.pdf
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The Kovacic algorithm and its improvements give explicit formulae for
the Liouvillian solutions of second order linear differential
equations. Algorithms for third order differential equations also
exist, but the tools they use are more sophisticated and the
computations more involved. In this paper we refine parts of the
algorithm to find Liouvillian solutions of third order equations. We
show that,except for four finite groups and a reduction to the second
order case, it is possible to give a formula in the imprimitve
case. We also give necessary conditions and several simplifications
for the computation of the minimal polynomial for the remaining finite
set of finite groups (or any known finite group) by extracting
ramification information from the character table. Several examples
have been constructed, illustrating the possibilities and limitations.
\end{adjustwidth}

\subsection{Expression Simplification} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Carette 04]{Car04} Carette, Jacques
``Understanding Expression Simplification''
\verbwww.cas.mcmaster.ca/~carette/publications/simplification.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Car04.pdf
+\bibitem[Minc 79]{Min79} Henryk Minc
+``Evaluation of Permanents''
+Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 2732.
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We give the first formal definition of the concept of {\sl
simplification} for general expressions in the context of Computer
Algebra Systems. The main mathematical tool is an adaptation of the
theory of Minimum Description Length, which is closely related to
various theories of complexity, such as Kolmogorov Complexity and
Algorithmic Information Theory. In particular, we show how this theory
can justify the use of various ``magic constants'' for deciding
between some equivalent representations of an expression, as found in
implementations of simplification routines.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[More 74]{MGH74} More J J.; Garbow B S.; Hillstrom K E.
+``User Guide for Minpack1''
+ANL8074 Argonne National Laboratory. (1974)
\subsection{Integration} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Adamchik xx]{Adamxx} Adamchik, Victor
``Definite Integration''
\verbwww.cs.cmu.edu/~adamchik/articles/integr/mj.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Adamxx.pdf
+\bibitem[Mikhlin 67]{MS67} Mikhlin S G.; Smolitsky K L.
+``Approximate Methods for the Solution of Differential and
+Integral Equations''
+Elsevier. (1967)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Adamchik 97]{Adam97} Adamchik, Victor
``A Class of Logarithmic Integrals''
\verbwww.cs.cmu.edu/~adamchik/articles/issac/issac97.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Adam97.pdf
+\bibitem[Mitchell 80]{MG80} Mitchell A R.; Griffiths D F.
+``The Finite Difference Method in Partial Differential Equations''
+Wiley. (1980)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A class of definite integrals involving cyclotomic polynomials and
nested logarithms is considered. The results are given in terms of
derivatives of the Hurwitz Zeta function. Some special cases for which
such derivatives can be expressed in closed form are also considered.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Avgoustis 77]{Avgo77} Avgoustis, Ioannis Dimitrios
``Definite Integration using the Generalized Hypergeometric Functions''
\verbdspace.mit.edu/handle/1721.1/16269
%\verbaxiomdeveloper.org/axiomwebsitep/papers/Avgo77.pdf
+\bibitem[Moler 73]{MS73} Moler C B.; Stewart G W.
+``An Algorithm for Generalized Matrix Eigenproblems''
+SIAM J. Numer. Anal. 10 241256. 1973
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A design for the definite integration of approximately fifty Special
Functions is described. The Generalized Hypergeometric Functions are
utilized as a basis for the representation of the members of the above
set of Special Functions. Only a relatively small number of formulas
that generally involve Generalized Hypergeometric Functions are
utilized for the integration stage. A last and crucial stage is
required for the integration process: the reduction of the Generalized
Hypergeometric Function to Elementary and/or Special Functions.
+\begin{chunk}{axiom.bib}
+@article{Muld97,
+ author = "Mulders, Thom",
+ title = "A Note on Subresultants and the Lazard/Rioboo/Trager Formula in
+ Rational Function Integration",
+ journal = "Journal of Symbolic Computation",
+ year = "1997",
+ volume = "24",
+ number = "1",
+ month = "July",
+ pages = "4550",
+ paper = "Muld97.pdf",
+ abstract = "
+ An ambiguity in a formula of Lazard, Rioboo and Trager, connecting
+ subresultants and rational function integration, is indicated and
+ examples of incorrect interpretations are given."
+}
The result of an early implementation which involves Laplace
transforms are given and some actual examples with their corresponding
timing are provided.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Baddoura 89]{Bad89} Baddoura, Jamil
``A Dilogarithmic Extension of Liouville's Theorem on Integration in Finite Terms''
\verbwww.dtic.mil/dtic/tr/fulltext/u2/a206681.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bad89.pdf
+\bibitem[Munksgaard 80]{Mun80} Munksgaard N.
+``Solving Sparse Symmetric Sets of Linear Equations by Preconditioned
+Conjugate Gradients''
+ACM Trans. Math. Softw. 6 206219. (1980)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The result obtained generalizes Liouville's Theorem by allowing, in
addition to the elementary functions, dilogarithms to appear in the
integral of an elementary function. The basic conclusion is that an
associated function to the dilogarihm, if dilogarithms appear in the
integral, appears linearly, with logarithms appearing in a nonlinear
way.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Baddoura 94]{Bad94} Baddoura, Mohamed Jamil
``Integration in Finite Terms with Elementary Functions and Dilogarithms''
\verbdspace.mit.edu/bitstream/handle/1721.1/26864/30757785.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bad94.pdf
+\bibitem[Murray 72]{Mur72} Murray W, (ed)
+``Numerical Methods for Unconstrained Optimization''
+Academic Press. (1972)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this thesis, we report on a new theorem that generalizes
Liouville's theorem on integration in finite terms. The new theorem
allows dilogarithms to occur in the integral in addition to elementary
functions. The proof is base on two identities for the dilogarithm,
that characterize all the possible algebraic relations among
dilogarithms of functions that are built up from the rational
functions by taking transcendental exponentials, dilogarithms, and
logarithms.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Baddoura 10]{Bad10} Baddoura, Jamil
``A Note on Symbolic Integration with Polylogarithms''
J. Math Vol 8 pp229241 (2011)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bad10.pdf
+\bibitem[Murtagh 83]{MS83} Murtagh B A.; Saunders M A
+``MINOS 5.0 User's Guide''
+Report SOL 8320. Department of Operations Research, Stanford University 1983
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We generalize partially Liouville's theorem on integration in finite
terms to allow polylogarithms of any order to occur in the integral in
addition to elementary functions. The result is a partial
generalization of a theorem proved by the author for the
dilogarithm. It is also a partial proof of a conjecture postulated by
the author in 1994. The basic conclusion is that an associated
function to the nth polylogarithm appears linearly with logarithms
appearing possibly in a polynomial way with nonconstant coefficients.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bajpai 70]{Bajp70} Bajpai, S.D.
``A contour integral involving legendre polynomial and Meijer's Gfunction''
\verblink.springer.com/article/10.1007/BF03049565
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bajp70.pdf
+\bibitem[Musser 78]{Mus78} Musser, David R.
+``On the Efficiency of a Polynomial Irreducibility Test''
+Journal of the ACM, Vol. 25, No. 2, April 1978, pp. 271282
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this paper a countour integral involving Legendre polynomial and
Meijer's Gfunction is evaluated. the integral is of general character
and it is a generalization of results recently given by Meijer,
MacRobert and others. An integral involving regular radial Coulomb
wave function is also obtained as a particular case.
\end{adjustwidth}
+\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Bronstein 89]{Bro89a} Bronstein, M.
``An Algorithm for the Integration of Elementary Functions''
Lecture Notes in Computer Science Vol 378 pp491497 (1989)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro89a.pdf
+\bibitem[Nijenhuis 78]{NW78} Nijenhuis and Wilf
+``Combinatorical Algorithms''
+Academic Press, New York 1978.
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Trager (1984) recently gave a new algorithm for the indefinite
integration of algebraic functions. His approach was ``rational'' in
the sense that the only algebraic extension computed in the smallest
one necessary to express the answer. We outline a generalization of
this approach that allows us to integrate mixed elementary
functions. Using only rational techniques, we are able to normalize
the integrand, and to check a necessary condition for elementary
integrability.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Bronstein 90a]{Bro90a} Bronstein, Manuel
``Integration of Elementary Functions''
J. Symbolic Computation (1990) 9, pp117173 September 1988
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro90a.pdf
+\bibitem[Nikolai 79]{Nik79} Nikolai P J.
+``Algorithm 538: Eigenvectors and eigenvalues of real generalized
+symmetric matrices by simultaneous iteration''
+ACM Trans. Math. Softw. 5 118125. (1979)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We extend a recent algorithm of Trager to a decision procedure for the
indefinite integration of elementary functions. We can express the
integral as an elementary function or prove that it is not
elementary. We show that if the problem of integration in finite terms
is solvable on a given elementary function field $k$, then it is
solvable in any algebraic extension of $k(\theta)$, where $\theta$ is
a logarithm or exponential of an element of $k$. Our proof considers
an element of such an extension field to be an algebraic function of
one variable over $k$.

In his algorithm for the integration of algebraic functions, Trager
describes a Hermitetype reduction to reduce the problem to an
integrand with only simple finite poles on the associated Riemann
surface. We generalize that technique to curves over liouvillian
ground fields, and use it to simplify our integrands. Once the
multipe finite poles have been removed, we use the Puiseux expansions
of the integrand at infinity and a generalization of the residues to
compute the integral. We also generalize a result of Rothstein that
gives us a necessary condition for elementary integrability, and
provide examples of its use.
\end{adjustwidth}
+\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@article{Bron90c,
 author = "Bronstein, Manuel",
 title = "On the integration of elementary functions",
 journal = "Journal of Symbolic Computation",
 volume = "9",
 number = "2",
 pages = "117173",
 year = "1990",
 month = "February"
+@misc{OCAM14,
+ author = "unknown",
+ title = "The OCAML website",
+ url = "http://ocaml.org"
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 93]{REFBS93} Bronstein, Manuel; Salvy, Bruno
``Full partial fraction decomposition of rational functions''
In Bronstein [Bro93] pp157160 ISBN 0897916042 LCCN QA76.95 I59 1993
\verbwww.acm.org/pubs/citations/proceedings/issac/164081/
+\bibitem[Ollagnier 94]{Olla94} Ollagnier, Jean Moulin
+``Algorithms and methods in differential algebra''
+\verbwww.lix.polytechnique.fr/~moulin/papiers/atelier.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Olla94.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 90]{Bro90b} Bronstein, Manuel
``A Unification of Liouvillian Extensions''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro90b.pdf
+\bibitem[Olver 10]{NIST10} Olver, Frank W.; Lozier, Daniel W.;
+Boisvert, Ronald F.; Clark, Charles W. (ed)
+``NIST Handbook of Mathematical Functions''
+(2010) Cambridge University Press ISBN 9780521192255
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We generalize Liouville's theory of elementary functions to a larger
class of differential extensions. Elementary, Liouvillian and
trigonometric extensions are all special cases of our extensions. In
the transcendental case, we show how the rational techniques of
integration theory can be applied to our extensions, and we give a
unified presentation which does not require separate cases for
different monomials.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@book{Bron97,
 author = "Bronstein, Manuel",
 title = "Symbolic Integration ITranscendental Functions",
 publisher = "Springer, Heidelberg",
 year = "1997",
 isbn = "3540214933",
 url = "http://evilwire.org/arrrXiv/Mathematics/Bronstein,_Symbolic_Integration_I,1997.pdf",
 paper = "Bron97.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[OpenM]{OpenM}.
+``OpenMath Technical Overview''
+\verbwww.openmath.org/overview/technical.html
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Bronstein 05a]{Bro05a} Bronstein, Manuel
``The Poor Man's Integrator, a parallel integration heuristic''
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/pmint/pmint.txt
\verbwwwsop.inria.fr/cafe/Manuel.Bronstein/pmint/examples
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro05a.txt
+\bibitem[Ortega 70]{OR70} Ortega J M.; Rheinboldt W C.
+``Iterative Solution of Nonlinear Equations in Several Variables''
+Academic Press. (1970)
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Bron06,
 author = "Bronstein, M.",
 title = "Parallel integration",
 journal = "Programming and Computer Software",
 year = "2006",
 issn = "03617688",
 volume = "32",
 number = "1",
 doi = "10.1134/S0361768806010075",
 url = "http://dx.doi.org/10.1134/S0361768806010075",
 publisher = "Nauka/Interperiodica",
 pages = "5960",
 paper = "Bron06.pdf"
+@misc{Ostr1845,
+ author = "Ostrogradsky. M.W.",
+ title = "De l'int\'{e}gration des fractions rationelles.",
+ journal = "Bulletin de la Classe PhysicoMath\'{e}matiques de
+ l'Acae\'{e}mie Imp\'{e}riale des Sciences de St. P\'{e}tersbourg,",
+ volume = "IV",
+ pages = "145167,286300",
+ year = "1845"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Parallel integration is an alternative method for symbolic
integration. While also based on Liouville's theorem, it handles all
the generators of the differential field containing the integrand ``in
parallel'', i.e. all at once rather than considering only the topmost
one in a recursive fasion. Although it still contains heuristic
aspects, its ease of implementation, speed, high rate of success, and
ability to integrate functions that cannot be handled by the Risch
algorithm make it an attractive alternative.
\end{adjustwidth}
+\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@article{Bron07,
 author = "Bronstein, Manuel",
 title = "Structure theorems for parallel integration",
 journal = "Journal of Symbolic Computation",
 volume = "42",
 number = "7",
 pages = "757769",
 year = "2007",
 month = "July",
 paper = "Bron07.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Paige 75]{PS75} Paige C C.; Saunders M A.
+``Solution of Sparse Indefinite Systems of Linear Equations''
+SIAM J. Numer. Anal. 12 617629. (1975)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We introduce structure theorems that refine Liouville's Theorem on
integration in closed form for general derivations on multivariate
rational function fields. By predicting the arguments of the new
logarithms that an appear in integrals, as well as the denominator of
the rational part, those theorems provide theoretical backing for the
RischNorman integration method. They also generalize its applicability
to nonmonomial extensions, for example the Lambert W function.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Charlwood 07]{Charl07} Charlwood, Kevin
``Integration on Computer Algebra Systems''
The Electronic J of Math. and Tech. Vol 2, No 3, ISSN 19332823
\verb12000.org/my_notes/ten_hard_integrals/paper.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Charl07.pdf
+\bibitem[Paige 82a]{PS82a} Paige C C.; Saunders M A.
+``LSQR: An Algorithm for Sparse Linear Equations and Sparse Leastsquares''
+ACM Trans. Math. Softw. 8 4371. (1982)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this article, we consider ten indefinite integrals and the ability
of three computer algebra systems (CAS) to evaluate them in
closedform, appealing only to the class of real, elementary
functions. Although these systems have been widely available for many
years and have undergone major enhancements in new versions, it is
interesting to note that there are still indefinite integrals that
escape the capacity of these systems to provide antiderivatves. When
this occurs, we consider what a user may do to find a solution with
the aid of a CAS.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Paige 82b]{PS82b} Paige C C.; Saunders M A.
+``ALGORITHM 583 LSQR: Sparse Linear Equations and Leastsquares Problems''
+ACM Trans. Math. Softw. 8 195209. (1982)
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Charlwood 08]{Charl08} Charlwood, Kevin
``Symbolic Integration Problems''
\verbwww.apmaths.uwo.ca/~arich/IndependentTestResults/CharlwoodIntegrationProblems.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Charl08.pdf
+\bibitem[Parker 84]{Par84} Parker, R. A.
+``The Computer Calculation of Modular Characters (The MeatAxe)''
+M. D. Atkinson (Ed.), Computational Group Theory
+Academic Press, Inc., London 1984
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A list of the 50 example integration problems from Kevin Charlwood's 2008
article ``Integration on Computer Algebra Systems''. Each integral along
with its optimal antiderivative (that is, the best antiderivative found
so far) is shown.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Parlett 80]{Par80} Parlett B N.
+``The Symmetric Eigenvalue Problem''
+PrenticeHall. 1980
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cherry 84]{Che84} Cherry, G.W.
``Integration in Finite Terms with Special Functions: The Error Function''
J. Symbolic Computation (1985) Vol 1 pp283302
%\verbaxiomdeveloper.org/axiomwebsite/papers/Che84.pdf
+\bibitem[Parnas 10]{PJ10} Parnas, David Lorge; Jin, Ying
+``Defining the meaning of tabular mathematical expressions''
+Science of Computer Programming V75 No.11 Nov 2010 pp9801000 Elesevier
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A decision procedure for integrating a class of transcendental
elementary functions in terms of elementary functions and error
functions is described. The procedure consists of three mutually
exclusive cases. In the first two cases a generalised procedure for
completing squares is used to limit the error functions which can
appear in the integral of a finite number. This reduces the problem
to the solution of a differential equation and we use a result of
Risch (1969) to solve it. The third case can be reduced to the
determination of what we have termed $\sum$decompositions. The resutl
presented here is the key procuedure to a more general algorithm which
is described fully in Cherry (1983).
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Parnas 95]{PM95} Parnas, David Lorge; Madey, Jan
+``Functional Documents for Computer Systems''
+Science of Computer Programming V25 No.1 Oct 1995 pp4161 Elesevier
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cherry 86]{Che86} Cherry, G.W.
``Integration in Finite Terms with Special Functions:
The Logarithmic Integral''
SIAM J. Comput. Vol 15 pp121 February 1986
+\bibitem[Paul 81]{Paul81} Paul, Richard
+``Robot Manipulators''
+MIT Press 1981
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@book{Pear56,
+ author = "Pearcey, T.",
+ title = "Table of the Fresnel Integral",
+ publisher = "Cambridge University Press",
+ year = "1956"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Cherry 89]{Che89} Cherry, G.W.
``An Analysis of the Rational Exponential Integral''
SIAM J. Computing Vol 18 pp 893905 (1989)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Che89.pdf
+\bibitem[Pereyra 79]{Per79} Pereyra V.
+``PASVA3: An Adaptive FiniteDifference Fortran Program for First Order
+Nonlinear, Ordinary Boundary Problems''
+Codes for Boundary Value Problems in Ordinary Differential Equations.
+Lecture Notes in Computer Science.
+(ed B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76
+SpringerVerlag. (1979)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this paper an algorithm is presented for integrating expressions of
the form $\int{ge^f~dx}$, where $f$ and $g$ are rational functions of
$x$, in terms of a class of special functions called the special
incomplete $\Gamma$ functions. This class of special functions
includes the exponential integral, the error functions, the sine and
cosing integrals, and the Fresnel integrals. The algorithm presented
here is an improvement over those published previously for integrating
with special functions in the following ways: (i) This algorithm
combines all the above special functions into one algorithm, whereas
previously they were treated separately, (ii) Previous algorithms
require that the underlying field of constants be algebraically
closed. This algorithm, however, works over any field of
characteristic zero in which the basic field operations can be carried
out. (iii) This algorithm does not rely on Risch's solution of the
differential equation $y^\prime + fy = g$. Instead, a more direct
method of undetermined coefficients is used.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Churchill 06]{Chur06} Churchill, R.C.
``Liouville's Theorem on Integration Terms of Elementary Functions''
\verbwww.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Chur06.pdf
+\bibitem[Peters 67a]{Pet67a} Peters G.
+``NPL Algorithms Library''
+Document No. F2/03/A. (1967)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This talk should be regarded as an elementary introduction to
differential algebra. It culminates in a purely algebraic proof, due
to M. Rosenlicht, of an 1835 theorem of Liouville on the existence of
``elementary'' integrals of ``elementary'' functions. The precise
meaning of elementary will be specified. As an application of that
theorem we prove that the indefinite integral $\int{e^{x^2}}~dx$
cannot be expressed in terms of elementary functions.
\begin{itemize}
\item Preliminaries on Meromorphic Functions
\item Basic (Ordinary) Differential Algebra
\item Differential Ring Extensions with No New Constants
\item Extending Derivations
\item Integration in Finite Terms
\end{itemize}
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Davenport 79b]{Dav79b} Davenport, James Harold
``On the Integration of Algebraic Functions''
SpringerVerlag Lecture Notes in Computer Science 102
ISBN 0387102906
+\bibitem[Peters 67b]{Pet67b} Peters G.
+``NPL Algorithms Library''
+Document No.F1/04/A (1967)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 79c]{Dav79c} Davenport, J. H.
``Algorithms for the Integration of Algebraic Functions''
Lecture Notes in Computer Science V 72 pp415425 (1979)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav79c.pdf
+\bibitem[Peters 70]{PW70} Peters G.; Wilkinson J H.
+``The Leastsquares Problem and Pseudoinverses''
+Comput. J. 13 309316. (1970)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The problem of finding elementary integrals of algebraic functions has
long been recognized as difficult, and has sometimes been thought
insoluble. Risch stated a theorem characterising the integrands with
elementary integrals, and we can use the language of algebraic
geometry and the techniques of Davenport to yield an algorithm that will
always produce the integral if it exists. We explain the difficulty in
the way of extending this algorithm, and outline some ways of solving
it. Using work of Manin we are able to solve the problem in all cases
where the algebraic expressions depend on a parameter as well as on
the variable of integration.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Davenport 82a]{Dav82a} Davenport, J.H.
``The Parallel Risch Algorithm (I)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav82a.pdf
+\bibitem[Peters 71]{PW71} Peters G.; Wilkinson J H.
+``Practical Problems Arising in the Solution of Polynomial Equations''
+J. Inst. Maths Applics. 8 1635. (1971)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this paper we review the socalled ``parallel Risch'' algorithm for
the integration of transcendental functions, and explain what the
problems with it are. We prove a positive result in the case of
logarithmic integrands.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Davenport 82]{Dav82} Davenport, J.H.
``On the Parallel Risch Algorithm (III): Use of Tangents''
SIGSAM V16 no. 3 pp36 August 1982
+\bibitem[Pierce 82]{Pie82} R.S. Pierce
+``Associative Algebras''
+Graduate Texts in Mathematics 88
+SpringerVerlag, Heidelberg, 1982, ISBN 0387906932
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 03]{Dav03} Davenport, James H.
``The Difficulties of Definite Integration''
\verbwww.researchgate.net/publication/
\verb247837653_The_Diculties_of_Definite_Integration/file/72e7e52a9b1f06e196.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav03.pdf
+\bibitem[Piessens 73]{Pie73} Piessens R.
+``An Algorithm for Automatic Integration''
+Angewandte Informatik. 15 399401. (1973)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Indefinite integration is the inverse operation to differentiation,
and, before we can understand what we mean by indefinite integration,
we need to understand what we mean by differentiation.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Fateman 02]{Fat02} Fateman, Richard
``Symbolic Integration''
\verbinst.eecs.berkeley.edu/~cs282/sp02/lects/14.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Fat02.pdf
+\bibitem[Piessens 74]{PMB74} Piessens R.;; Mertens I.; Branders M.
+``Integration of Functions having Endpoint Singularities''
+Angewandte Informatik. 16 6568. (1974)
\end{chunk}
\begin{chunk}{axiom.bib}
@inproceedings{Gedd89,
 author = "Geddes, K. O. and Stefanus, L. Y.",
 title = "On the Rischnorman Integration Method and Its Implementation in MAPLE",
 booktitle = "Proceedings of the ACMSIGSAM 1989 International Symposium on Symbolic and Algebraic Computation",
 series = "ISSAC '89",
 year = "1989",
 isbn = "0897913256",
 location = "Portland, Oregon, USA",
 pages = "212217",
 numpages = "6",
 url = "http://doi.acm.org/10.1145/74540.74567",
 doi = "10.1145/74540.74567",
 acmid = "74567",
 publisher = "ACM",
 address = "New York, NY, USA",
 paper = "Gedd89.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Piessens 75]{PB75} Piessens R.; Branders M.
+``Algorithm 002. Computation of Oscillating Integrals''
+J. Comput. Appl. Math. 1 153164. (1975)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Unlike the Recursive Risch Algorithm for the integration of
transcendental elementary functions, the RischNorman Method processes
the tower of field extensions directly in one step. In addition to
logarithmic and exponential field extensions, this method can handle
extentions in terms of tangents. Consequently, it allows trigonometric
functions to be treated without converting them to complex exponential
form. We review this method and describe its implementation in
MAPLE. A heuristic enhancement to this method is also presented.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Geddes 92a]{GCL92a} Geddes, K.O.; Czapor, S.R.; Labahn, G.
``The Risch Integration Algorithm''
Algorithms for Computer Algebra, Ch 12 pp511573 (1992)
%\verbaxiomdeveloper.org/axiomwebsite/papers/GCL92a.pdf
+\bibitem[Piessens 76]{PVRBM76} Piessens R.; Van RoyBranders M.; Mertens I.
+``The Automatic Evaluation of Cauchy Principal Value Integrals''
+Angewandte Informatik. 18 3135. (1976)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Hardy 1916]{Hard16} Hardy, G.H.
``The Integration of Functions of a Single Variable''
Cambridge Unversity Press, Cambridge, 1916
% REF:00002
+\bibitem[Piessens 83]{PDUK83} Piessens R.; De DonckerKapenga E.;
+Uberhuber C.; Kahaner D.
+``QUADPACK, A Subroutine Package for Automatic Integration''
+SpringerVerlag.(1983)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Harrington 78]{Harr87} Harrington, S.J.
``A new symbolic integration system in reduce''
\verbcomjnl.oxfordjournals.or/content/22/2/127.full.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Harr87.pdf
+\bibitem[Polya 37]{Pol37} Polya, G.
+``Kombinatorische Anzahlbestimmungen fur Gruppen,
+Graphen und chemische Verbindungen''
+Acta Math. 68 (1937) 145254.
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A new integration system, employing both algorithmic and pattern match
integration schemes is presented. The organization of the system
differs from that of earlier programs in its emphasis on the
algorithmic approach to integration, its modularity and its ease of
revision. The new NormanRish algorithm and its implementation at the
University of Cambridge are employed, supplemented by a powerful
collection of simplification and transformation rules. The facility
for user defined integrals and functions is also included. The program
is both fast and powerful, and can be easily modified to incorporate
anticipated developments in symbolic integration.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@misc{Herm1872,
 author = "Hermite, E.",
 title = "Sur l'int\'{e}gration des fractions rationelles",
 journal = "Nouvelles Annales de Math\'{e}matiques",
 volume = "11",
 pages = "145148",
 year = "1872"
}
+\begin{chunk}{ignore}
+\bibitem[Powell 70]{Pow70} Powell M J D.
+``A Hybrid Method for Nonlinear Algebraic Equations''
+Numerical Methods for Nonlinear Algebraic Equations.
+(ed P Rabinowitz) Gordon and Breach. (1970)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Horowitz 71]{Horo71} Horowitz, Ellis
``Algorithms for Partial Fraction Decomposition and Rational Function Integration''
SYMSAC '71 Proc. ACM Symp. on Symbolic and Algebraic Manipulation (1971)
pp441457
%\verbaxiomdeveloper.org/axiomwebsite/papers/Horo71.pdf REF:00018
+\bibitem[Powell 74]{Pow74} Powell M J D.
+``Introduction to Constrained Optimization''
+Numerical Methods for Constrained Optimization.
+(ed P E Gill and W Murray) Academic Press. pp128. 1974
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Algorithms for symbolic partial fraction decomposition and indefinite
integration of rational functions are described. Two types of
partial fraction decomposition are investigated, squarefree and
complete squarefree. A method is derived, based on the solution of
a linear system, which produces the squarefree decomposition of any
rational function, say A/B. The computing time is show to be
$O(n^4(ln nf)^2)$ where ${\rm deg}(A) < {\rm\ deg}(B) = n$ and $f$
is a number which is closely related to the size of the coefficients
which occur in A and B. The complete squarefree partical fraction
decomposition can then be directly obtained and it is shown that the
computing time for this process is also bounded by $O(n^4(ln nf)^2)$.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Jeffrey 97]{Jeff97} Jeffrey, D.J.; Rich, A.D.
``Recursive integration of piecewisecontinuous functions''
\verbwww.cybertester.com/data/recint.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Jeff97.pdf
+\bibitem[Powell 83]{Pow83} Powell M J D.
+``Variable Metric Methods in Constrained Optimization''
+Mathematical Programming: The State of the Art.
+(ed A Bachem, M Groetschel and B Korte) SpringerVerlag. pp288311. 1983
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
An algorithm is given for the integration of a class of
piecewisecontinuous functions. The integration is with respect to a
real variable, because the functions considered do not in general
allow integration in the complex plane to be defined. The class of
integrands includes commonly occurring waveforms, such as square
waves, triangular waves, and the floor function; it also includes the
signum function. The algorithm can be implemented recursively, and it
has the property of ensuring that integrals are continuous on domains
of maximum extent.
\end{adjustwidth}
+\begin{chunk}{axiom.bib}
+@inproceedings{Prat73,
+ author = "Pratt, Vaughan R.",
+ title = "Top down operator precedence",
+ booktitle = "Proc. 1st annual ACM SIGACTSIGPLAN Symposium on Principles
+ of Programming Languages",
+ series = "POPL'73",
+ pages = "4151",
+ year = "1973",
+ url = "http://hall.org.ua/halls/wizzard/pdf/Vaughan.Pratt.TDOP.pdf",
+ keywords = "axiomref",
+ paper = "Prat73.pdf"
+}
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jeffrey 99]{Jeff99} Jeffrey, D.J.; Labahn, G.; Mohrenschildt, M.v.;
Rich, A.D.
``Integration of the signum, piecewise and related functions''
\verbcs.uwaterloo.ca/~glabahn/Papers/issac992.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Jeff99.pdf
+\bibitem[Press 95]{PTVF95} Press, William H.; Teukolsky, Saul A.;
+Vetterling, William T.; Flannery, Brian P.
+``Numerical Recipes in C''
+Cambridge University Press (1995) ISBN 0521431085
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
When a computer algebra system has an assumption facility, it is
possible to distinguish between integration problems with respect to a
real variable, and those with respect to a complex variable. Here, a
class of integration problems is defined in which the integrand
consists of compositions of continuous functions and signum functions,
and integration is with respect to a real variable. Algorithms are
given for evaluating such integrals.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Kiymaz 04]{Kiym04} Kiymaz, Onur; Mirasyedioglu, Seref
``A new symbolic computation for formal integration with exact power series''
%\verbaxiomdeveloper.org/axiomwebsite/Kiym04.pdf
+\bibitem[Pryce 77]{PH77} Pryce J D.; Hargrave B A.
+``The Scale Pruefer Method for oneparameter and multiparameter eigenvalue
+problems in ODEs''
+Inst. Math. Appl., Numerical Analysis Newsletter. 1(3) (1977)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper describes a new symbolic algorithm for formal integration
of a class of functions in the context of exact power series by using
generalized hypergeometric series and computer algebraic technique.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Knowles 93]{Know93} Knowles, P.
``Integration of a class of transcendental liouvillian
functions with errorfunctions i''
Journal of Symbolic Computation Vol 13 pp525543 (1993)
+\bibitem[Pryce 81]{Pry81} Pryce J D.
+``Two codes for SturmLiouville problems''
+Technical Report CS8101. Dept of Computer Science, Bristol University (1981)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Knowles 95]{Know95} Knowles, P.
``Integration of a class of transcendental liouvillian
functions with errorfunctions ii''
Journal of Symbolic Computation Vol 16 pp227241 (1995)
+\bibitem[Pryce 86]{Pry86} Pryce J D.
+``Error Estimation for Phasefunction Shooting Methods for
+SturmLiouville Problems''
+J. Num. Anal. 6 103123. (1986)
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Krag09,
 author = "Kragler, R.",
 title = "On Mathematica Program for Poor Man's Integrator Algorithm",
 journal = "Programming and Computer Software",
 volume = "35",
 number = "2",
 pages = "6378",
 year = "2009",
 issn = "03617688",
 paper = "Krag09.pdf"
+@misc{Puff09,
+ author = "Puffinware LLC",
+ title = "Singular Value Decomposition (SVD) Tutorial",
+ url = "http://www.puffinwarellc.com/p3a.htm"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this paper by means of computer experiment we study advantages and
disadvantages of the heuristical method of ``parallel integrator''. For
this purpose we describe and use implementation of the method in
Mathematica. In some cases we compare this implementation with the original
one in Maple.
\end{adjustwidth}
+\subsection{Q} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Lang 93]{Lang93} Lang, S.
``Algebra''
AddisonWesly, New York, 3rd edition 1993
+\bibitem[QuintanaOrti 06]{QG06} QuintanaOrti, Gregorio;
+van de Geijn, Robert
+``Improving the performance of reduction to Hessenberg form''
+ACM Transactions on Mathematical Software, 32(2):180194, June 2006.
\end{chunk}
+\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Leerawat 02]{Leer02} Leerawat, Utsanee; Laohakosol, Vichian
``A Generalization of Liouville's Theorem on Integration in Finite Terms''
\verbwww.mathnet.or.kr/mathnet/kms_tex/113666.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Leer02.pdf
+\bibitem[Rabinowitz 70]{Rab70} Rabinowitz P.
+``Numerical Methods for Nonlinear Algebraic Equations''
+Gordon and Breach. (1970)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
A generalization of Liouville's theorem on integration in finite
terms, by enlarging the class of fields to an extension called
EiGamma extension is established. This extension includes the
$\mathcal{E}\mathcal{L}$elementary extensions of Singer, Saunders and
Caviness and contains the Gamma function.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Leslie 09]{Lesl09} Leslie, Martin
``Why you can't integrate exp($x^2$)''
\verbmath.arizona.edu/~mleslie/files/integrationtalk.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Lesl09.pdf
+\bibitem[Ralston 65]{Ral65} Ralston A.
+``A First Course in Numerical Analysis''
+McGrawHill. 8790. (1965)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Lichtblau 11]{Lich11} Lichtblau, Daniel
``Symbolic definite (and indefinite) integration: methods and open issues''
ACM Comm. in Computer Algebra Issue 175, Vol 45, No.1 (2011)
\verbwww.sigsam.org/bulletin/articles/175/issue175.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Lich11.pdf
+\bibitem[Ramakrishnan 03]{Ram03} Ramakrishnan, Maya
+``A Gentle Introduction to Lyapunov Functions''
+ORSUM August 2003
+\verbwww.or.ms.unimelb.edu.au/handouts/lyaptalk.1.pdf
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The computation of definite integrals presents one with a variety of
choices. There are various methods such as NewtonLeibniz or Slater's
convolution method. There are questions such as whether to split or
merge sums, how to search for singularities on the path of
integration, when to issue conditional results, how to assess
(possibly conditional) convergence, and more. These various
considerations moreover interact with one another in a multitude of
ways. Herein we discuss these various issues and illustrate with examples.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Ramsey 03]{Ra03} Ramsey, Norman
+``NowebA Simple, Extensible Tool for Literate Programming''
+\verbwww.eecs.harvard.edu/~nr/noweb
\begin{chunk}{axiom.bib}
@article{Liou1833a,
 author = "Liouville, Joseph",
 title = "Premier m\'{e}moire sur la d\'{e}termination des int\'{e}grales dont la valeur est alg\'{e}brique",
 journal = "Journal de l'Ecole Polytechnique",
 volume = "14",
 pages = "124128",
 year = "1833"
}
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Redfield 27]{Red27} Redfield, J.H.
+``The Theory of GroupReduced Distributions''
+American J. Math., 49 (1927) 433455.
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Liou1833b,
 author = "Liouville, Joseph",
 title = "Second m\'{e}moire sur la d\'{e}termination des int\'{e}grales dont la valeur est alg\'{e}brique",
 journal = "Journal de l'Ecole Polytechnique",
 volume = "14",
 pages = "149193",
 year = "1833"
}
+\begin{chunk}{ignore}
+\bibitem[Reinsch 67]{Rei67} Reinsch C H.
+``Smoothing by Spline Functions''
+Num. Math. 10 177183. (1967)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Liouville 1833c]{Lio1833c} Liouville, Joseph
``Note sur la determination des int\'egrales dont la
valeur est alg\'ebrique''
Journal f\"ur die Reine und Angewandte Mathematik,
Vol 10 pp 247259, (1833)
+\bibitem[Renka 84]{Ren84} Renka R L.
+``Algorithm 624: Triangulation and Interpolation of Arbitrarily Distributed
+Points in the Plane''
+ACM Trans. Math. Softw. 10 440442. (1984)
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Renka 84]{RC84} Renka R L.; Cline A K.
+``A Trianglebased C Interpolation Method''
+Rocky Mountain J. Math. 14 223237. (1984)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Liouville 1833d]{Lio1833d} Liouville, Joseph
``Sur la determination des int\'egrales dont la valeur est
alg\'ebrique''
{\sl Journal de l'Ecole Polytechnique}, 14:124193, 1833
+\bibitem[Reutenauer 93]{Re93} Reutenauer, Christophe
+``Free Lie Algebras''
+Oxford University Press, June 1993 ISBN 0198536798
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Liouville 1835]{Lio1835} Liouville, Joseph
``M\'emoire sur l'int\'gration d'une classe de fonctions
transcendentes''
Journal f\"ur die Reine und Angewandte Mathematik,
Vol 13(2) pp 93118, (1835)
+\bibitem[Reznick 93]{Rezn93} Reznick, Bruce
+``An Inequality for Products of Polynomials''
+Proc. AMS Vol 117 No 4 April 1993
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Rezn93.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Marc 94]{Marc94} Marchisotto, Elena Anne; Zakeri, GholemAll
``An Invitation to Integration in Finite Terms''
College Mathematics Journal Vol 25 No 4 (1994) pp295308
\verbwww.rangevoting.org/MarchisottoZint.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Marc94.pdf
+\bibitem[Rich xx]{Rixx} Rich, A.D.; Jeffrey, D.J.
+``Crafting a Repository of Knowledge Based on Transformation''
+\verbwww.apmaths.uwo.ca/~djeffrey/Offprints/IntegrationRules.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Rixx.pdf
+ abstract = "
+ We describe the development of a repository of mathematical knowledge
+ based on transformation rules. The specific mathematical problem is
+ indefinite integration. It is important that the repository be not
+ confused with a lookup table. The database of transformation rules is
+ at present encoded in Mathematica, but this is only one convenient
+ form of the repository, and it could be readily translated into other
+ formats. The principles upon which the set of rules is compiled is
+ described. One important principle is minimality. The benefits of the
+ approach are illustrated with examples, and with the results of
+ comparisons with other approaches."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Marik 91]{Mari91} Marik, Jan
``A note on integration of rational functions''
\verbdml.cz/bitstream/handle/10338.dmlcz/126024/MathBohem_11619914_9.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Mari91.pdf
+\bibitem[Rich 10]{Ri10} Rich, Albert D.
+``Rulebased Mathematics''
+\verbwww.apmaths.uwo.ca/~arich
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Let $P$ and $Q$ be polynomials in one variable with complex coefficients
and let $n$ be a natural number. Suppose that $Q$ is not constant and
has only simple roots. Then there is a rational function $\varphi$
with $\varphi^\prime=P/Q^{n+1}$ if and only if the Wronskian of the
functions $Q^\prime$, $(Q^2)^\prime,\ldots\,(Q^n)^\prime$,$P$ is
divisible by $Q$.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Moses 76]{Mos76} Moses, Joel
``An introduction to the Risch Integration Algorithm''
ACM Proc. 1976 annual conference pp425428
%\verbaxiomdeveloper.org/axiomwebsite/papers/Mos76.pdf REF:00048
+\bibitem[Richardson 94]{RF94} Richardson, Dan; Fitch, John
+``The identity problem for elementary functions and constants''
+ACM Proc. of ISSAC 94 pp285290 ISBN 0897916387
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Risch's decision procedure for determining the integrability in closed
form of the elementary functions of the calculus is presented via
examples. The exponential and logarithmic cases of the algorithsm had
been implemented for the MACSYMA system several years ago. The
implementation of the algebraic case of the algorithm is the subject
of current research.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Moses 71a]{Mos71a} Moses, Joel
``Symbolic Integration: The Stormy Decade''
CACM Aug 1971 Vol 14 No 8 pp548560
\verbwwwinst.eecs.berkeley.edu/~cs282/sp02/readings/mosesint.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Mos71a.pdf REF:00017
+\bibitem[Richtmyer 67]{RM67} Richtmyer R D.; Morton K W.
+``Difference Methods for Initialvalue Problems''
+Interscience (2nd Edition). (1967)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Three approaches to symbolic integration in the 1960's are
described. The first, from artificial intelligence, led to Slagle's
SAINT and to a large degree to Moses' SIN. The second, from algebraic
manipulation, led to Monove's implementation and to Horowitz' and
Tobey's reexamination of the Hermite algorithm for integrating
rational functions. The third, from mathematics, led to Richardson's
proof of the unsolvability of the problem for a class of functions and
for Risch's decision procedure for the elementary functions.
Generalizations of Risch's algorithm to a class of special
functions and programs for solving differential equations and for
finding the definite integral are also described.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Norman 79]{Nor79} Norman, A.C.; Davenport, J.H.
``Symbolic Integration  The Dust Settles?''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Nor79.pdf
+\bibitem[Rioboo 92]{REFRio92} Rioboo, R.
+``Real algebraic closure of an ordered field, implementation in Axiom''
+In Wang [Wan92], pp206215 ISBN 0897914899 (soft cover)
+In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206215.
+0897914902 (hard cover) LCCN QA76.95.I59 1992
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
By the end of the 1960s it had been shown that a computer could find
indefinite integrals with a competence exceeding that of typical
undergraduates. This practical advance was backed up by algorithmic
interpretations of a number of clasical results on integration, and by
some significant mathematical extensions to these same results. At
that time it would have been possible to claim that all the major
barriers in the way of a complete system for automated analysis had
been breached. In this paper we survey the work that has grown out of
the abovementioned early results, showing where the development has
been smooth and where it has spurred work in seemingly unrelated fields.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Ostrowski 46]{Ost46} Ostrowski, A.
``Sur l'int\'egrabilit\'e \'el\'ementaire de quelques classes
d'expressions''
Comm. Math. Helv., Vol 18 pp 283308, (1946)
% REF:00008
+\bibitem[Rioboo 96]{Rio96} Rioboo, R.
+``Generic computation of the real closure of an ordered field''
+In Mathematics and Computers in Simulation Volume 42, Issue 46,
+November 1996.
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Raab 12]{Raab12} Raab, Clemens G.
``Definite Integration in Differential Fields''
\verbwww.risc.jku.at/publications/download/risc_4583/PhD_CGR.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Raab12.pdf
+\bibitem[Ritt 50]{Ritt50} Ritt, Joseph Fels
+``Differential Algebra''
+AMS Colloquium Publications Volume 33 ISBN 9780821846384
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The general goal of this thesis is to investigate and develop computer
algebra tools for the simplification resp. evaluation of definite
integrals. One way of finding the value of a def inite integral is
via the evaluation of an antiderivative of the integrand. In the
nineteenth century Joseph Liouville was among the first who analyzed
the structure of elementary antiderivatives of elementary functions
systematically. In the early twentieth century the algebraic structure
of differential fields was introduced for modeling the differential
properties of functions. Using this framework Robert H. Risch
published a complete algorithm for transcendental elementary
integrands in 1969. Since then this result has been extended to
certain other classes of integrands as well by Michael F. Singer,
Manuel Bronstein, and several others. On the other hand, if no
antiderivative of suitable form is available, then linear relations
that are satisfied by the parameter integral of interest may be found
based on the principle of parametric integration (often called
differentiating under the integral sign or creative telescoping).

The main result of this thesis extends the results mentioned above to
a complete algo rithm for parametric elementary integration for a
certain class of integrands covering a majority of the special
functions appearing in practice such as orthogonal polynomials,
polylogarithms, Bessel functions, etc. A general framework is provided
to model those functions in terms of suitable differential fields. If
the integrand is Liouvillian, then the present algorithm considerably
improves the efficiency of the corresponding algorithm given by Singer
et al. in 1985. Additionally, a generalization of Czichowskiâ€™s
algorithm for computing the logarithmic part of the integral is
presented. Moreover, also partial generalizations to include other
types of integrands are treated.

As subproblems of the integration algorithm one also has to find
solutions of linear or dinary differential equations of a certain
type. Some contributions are also made to solve those problems in our
setting, where the results directly dealing with systems of
differential equations have been joint work with Moulay A. Barkatou.

For the case of Liouvillian integrands we implemented the algorithm in
form of our Mathematica package Integrator. Parts of the
implementation also deal with more general functions. Our procedures
can be applied to a significant amount of the entries in integral
tables, both indefinite and definite integrals. In addition, our
procedures have been successfully applied to interesting examples of
integrals that do not appear in these tables or for which current
standard computer algebra systems like Mathematica or Maple do not
succeed. We also give examples of how parameter integrals coming from
the work of other researchers can be solved with the software, e.g.,
an integral arising in analyzing the entropy of certain processes.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Raab 13]{Raab13} Raab, Clemens G.
``Generalization of Risch's Algorithm to Special Functions''
\verbarxiv.org/pdf/1305.1481
%\verbaxiomdeveloper.org/axiomwebsite/papers/Raab13.pdf
+\bibitem[Rote 01]{Rote01} Rote, G\"unter
+``Divisionfree algorithms for the determinant and the Pfaffian''
+in Computational Discrete Mathematics ISBN 3540427759 pp119135
+\verbpage.mi.fuberlin.de/rote/Papers/pdf/Divisionfree+algorithms.pdf
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Symbolic integration deals with the evaluation of integrals in closed
form. We present an overview of Risch's algorithm including recent
developments. The algorithms discussed are suited for both indefinite
and definite integration. They can also be used to compute linear
relations among integrals and to find identities for special functions
given by parameter integrals. The aim of this presentation is twofold:
to introduce the reader to some basic idea of differential algebra in
the context of integration and to raise awareness in the physics
community of computer algebra algorithms for indefinite and definite
integration.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Raab xx]{Raabxx} Raab, Clemens G.
``Integration in finite terms for Liouvillian functions''
\verbwww.mmrc.iss.ac.cn/~dart4/posters/Raab.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Raabxx.pdf
+\bibitem[Rubey 07]{Rub07} Rubey, Martin
+``Formula Guessing with Axiom''
+April 2007
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Computing integrals is a common task in many areas of science,
antiderivatives are one way to accomplish this. The problem of
integration in finite terms can be states as follows. Given a
differential field $(F,D)$ and $f \in F$, compute $g$ in some
elementary extension of $(F,D)$ such that $Dg = f$ if such a $g$
exists.
+\begin{chunk}{ignore}
+\bibitem[Rutishauser 69]{Rut69} Rutishauser H.
+``Computational aspects of F L Bauer's simultaneous iteration method''
+Num. Math. 13 413. (1969)
This problem has been solved for various classes of fields $F$. For
rational functions $(C(x), \frac{d}{dx})$ such a $g$ always exists and
algorithms to compute it are known already for a long time. In 1969
Risch published an algorithm that solves this problem when $(F,D)$ is
a transcendental elementary extension of $(C(x),\frac{d}{dx})$. Later
this has been extended towards integrands being Liouvillian functions
by Singer et. al. via the use of regular logexplicit extensions of
$(C(x),\frac{d}{dx})$. Our algorithm extends this to handling
transcendental Liouvillian extensions $(F,D)$ of $(C,0)$ directly
without the need to embed them into logexplicit extensions. For
example, this means that
\[\int{(zx)x^{z1}e^{x}dx} = x^ze^{x}\]
can be computed without including log(x) in the differential field.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rich 09]{Rich09} Rich, A.D.; Jeffrey, D.J.
``A Knowledge Repository for Indefinite Integration Based on Transformation Rules''
\verbwww.apmaths.uwo.ca/~arich/A%2520Rulebased%2520Knowedge%2520Repository.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Rich09.pdf
+\bibitem[Rutishauser 70]{Rut70} Rutishauser H.
+``Simultaneous iteration method for symmetric matrices''
+Num. Math. 16 205223. (1970)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Taking the specific problem domain of indefinite integration, we
describe the ongoing development of a repository of mathematical
knowledge based on transformation rules. It is important that the
repository be not confused with a lookup table. The database of
transformation rules is at present encoded in Mathematica, but this is
only one convenient form of the repository, and it could be readily
translated into other formats. The principles upon which the set of
rules is compiled is described. One important principle is
minimality. The benefits of the approach are illustrated with
examples, and with the results of comparisons with other approaches.
\end{adjustwidth}
+\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@techreport{Risc68,
 author = "Risch, Robert",
 title = "On the integration of elementary functions which are built up using algebraic operations",
 type = "Research Report",
 number = "SP2801/002/00",
 institution = "System Development Corporation, Santa Monica, CA, USA",
 year = "1968"
}
+\begin{chunk}{ignore}
+\bibitem[Schafer 66]{Sch66} Schafer, R.D.
+``An Introduction to Nonassociative Algebras''
+Academic Press, New York, 1966
\end{chunk}
\begin{chunk}{axiom.bib}
@techreport{Risc69a,
 author = "Risch, Robert",
 title = "Further results on elementary functions",
 type = "Research Report",
 number = "RC2042",
 institution = "IBM Research, Yorktown Heights, NY, USA",
 year = "1969"

}
+\begin{chunk}{ignore}
+\bibitem[Schoenberg 53]{SW53} Schoenberg I J.; Whitney A.
+``On Polya Frequency Functions III''
+Trans. Amer. Math. Soc. 74 246259. (1953)
\end{chunk}
\begin{chunk}{axiom.bib}
@article{Risc69b,
 author = "Risch, Robert",
 title = "The problem of integration in finite terms",
 journal = "Transactions of the American Mathematical Society",
 volume = "139",
 year = "1969",
 pages = "167189",
 paper = "Ris69b.pdf",
 abstract = "This paper deals with the problem of telling whether a
 given elementary function, in the sense of analysis, has an elementary
 indefinite integral."

}
+\begin{chunk}{ignore}
+\bibitem[Schoenhage 82]{Sch82} Schoenhage, A.
+``The fundamental theorem of algebra in terms of computational complexity''
+preliminary report, Univ. Tuebingen, 1982
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper deals with the problem of telling whether a given elementary
function, in the sense of analysis, has an elementary indefinite integral.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{Risc70,
 author = "Risch, Robert",
 title = "The Solution of the Problem of Integration in Finite Terms",
 journal = "Bull. AMS",
 year = "1970",
 issn = "00029904",
 volume = "76",
 number = "3",
 pages = "605609",
 paper = "Risc70.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Schonfelder 76]{Sch76} Schonfelder J L.
+``The Production of Special Function Routines for a MultiMachine Library''
+Software Practice and Experience. 6(1) (1976)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The problem of integration in finite terms asks for an algorithm for
deciding whether an elementary function has an elementary indefinite
integral and for finding the integral if it does. ``Elementary'' is
used here to denote those functions build up from the rational
functions using only exponentiation, logarithms, trigonometric,
inverse trigonometric and algebraic operations. This vaguely worded
question has several precise, but inequivalent formulations. The
writer has devised an algorithm which solves the classical problem of
Liouville. A complete account is planned for a future publication. The
present note is intended to indiciate some of the ideas and techniques
involved.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{Risc79,
 author = "Risch, Robert",
 title = "Algebraic properties of the elementary functions of analysis",
 journal = "American Journal of Mathematics",
 volume = "101",
 pages = "743759",
 year = "1979"
+@book{Segg93,
+ author = "{von Seggern}, David Henry",
+ title = "CRC Standard Curves and Surfaces",
+ publisher = "CRC Press",
+ year = "1993",
+ isbn = "0849301963"
}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ritt 48]{Ritt48} Ritt, J.F.
``Integration in Finite Terms''
Columbia University Press, New York 1948
% REF:00046
+\bibitem[Seiler 95a]{Sei95a} Seiler, W.M.; Calmet, J.
+``JET  An Axiom Environment for Geometric Computations with Differential
+Equations''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Sei95a.pdf
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rosenlicht 68]{Ro68} Rosenlicht, Maxwell
``Liouville's Theorem on Functions with Elementary Integrals''
Pacific Journal of Mathematics Vol 24 No 1 (1968)
\verbmsp.org/pjm/1968/241/pjmv24n1p16p.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Ro68.pdf REF:00047
+\bibitem[Shepard 68]{She68} Shepard D.
+``A Twodimensional Interpolation Function for Irregularly Spaced Data''
+Proc. 23rd Nat. Conf. ACM. Brandon/Systems Press Inc.,
+Princeton. 517523. 1968
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Defining a function with one variable to be elemetary if it has an
explicit representation in terms of a finite number of algebraic
operations, logarithms, and exponentials. Liouville's theorem in its
simplest case says that if an algebraic function has an elementary
integral then the latter is itself an algebraic function plus a sum of
constant multiples of logarithms of algebraic functions. Ostrowski has
generalized Liouville's results to wider classes of meromorphic
functions on regions of the complex plane and J.F. Ritt has given the
classical account of the entire subject in his Integraion in Finite
Terms, Columbia University Press, 1948. In spite of the essentially
algebraic nature of the problem, all proofs so far have been analytic.
This paper gives a self contained purely algebraic exposition of the
probelm, making a few new points in addition to the resulting
simplicity and generalization.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{Rose72,
 author = "Rosenlicht, Maxwell",
 title = "Integration in finite terms",
 journal = "American Mathematical Monthly",
 year = "1972",
 volume = "79",
 pages = "963972",
 paper = "Rose72.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Shirayanagi 96]{Shir96} Shirayanagi, Kiyoshi
+``Floating point Gr\"obner bases''
+Mathematics and Computers in Simulation 42 pp 509528 (1996)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Shir96.pdf
+ abstract = "
+ Bracket coefficients for polynomials are introduced. These are like
+ specific precision floating point numbers together with error
+ terms. Working in terms of bracket coefficients, an algorithm that
+ computes a Gr{\"o}bner basis with floating point coefficients is
+ presented, and a new criterion for determining whether a bracket
+ coefficient is zero is proposed. Given a finite set $F$ of polynomials
+ with real coefficients, let $G_\mu$ be the result of the algorithm for
+ $F$ and a precision $\mu$, and $G$ be a true Gr{\"o}bner basis of
+ $F$. Then, as $\mu$ approaches infinity, $G_\mu$ converges to $G$
+ coefficientwise. Moreover, there is a precision $M$ such that if
+ $\mu \ge M$, then the sets of monomials with nonzero coefficients of
+ $G_\mu$ and $G$ are exactly the same. The practical usefulness of the
+ algorithm is suggested by experimental results."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rothstein 76]{Ro76} Rothstein, Michael
``Aspects of symbolic integration and simplifcation of exponential
and primitive functions''
PhD thesis, University of WisconsinMadison (1976)
\verbwww.cs.kent.edu/~rothstei/dis.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Ro76.pdf REF:00051
+\bibitem[Sims 71]{Sims71} Sims, C.
+``Determining the Conjugacy Classes of a Permutation Group''
+Computers in Algebra and Number Theory, SIAMAMS Proc., Vol. 4,
+American Math. Soc., 1991, pp191195
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this thesis we cover some aspects of the theory necessary to obtain
a canonical form for functions obtained by integration and
exponentiation from the set of rational functions.

These aspects include a new algorithm for symbolic integration of
functions involving logarithms and exponentials which avoids
factorization of polynomials in those cases where algebraic extension
of the constant field is not required, avoids partial fraction
decompositions, and only solves linear systems with a small number of
unknowns.

We have also found a theorem which states, roughly speaking, that if
integrals which can be represented as logarithms are represented as
such, the only algebraic dependence that a new exponential or
logarithm can satify is given by the law of exponents or the law of
logarithms.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Rothstein 76a]{Ro76a} Rothstein, Michael; Caviness, B.F.
``A structure theorem for exponential and primitive functions: a preliminary report''
ACM Sigsam Bulletin Vol 10 Issue 4 (1976)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Ro76a.pdf
+\bibitem[Singer 89]{Sing89} Singer, M.F.
+``Formal Solutions of Differential Equations''
+J. Symbolic COmputation 10, No.1 5994 (1990)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Sing89.pdf
+ keywords = "survey",
+ abstract = "
+ We give a survey of some methods for finding formal solutions of
+ differential equations. These include methods for finding power series
+ solutions, elementary and liouvillian solutions, first integrals, Lie
+ theoretic methods, transform methods, asymptotic methods. A brief
+ discussion of difference equations is also included."
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this paper a generalization of the Risch Structure Theorem is reported.
The generalization applies to fields $F(t_1,\ldots,t_n)$ where $F$
is a differential field (in our applications $F$ will be a finitely
generated extension of $Q$, the field of rational numbers) and each $t_i$
is either algebraic over $F_{i1}=F(t_1,\ldots,t_{i1})$, is an exponential
of an element in $F_{i1}$, or is an integral of an element in $F_{i1}$.
If $t_i$ is an integral and can be expressed using logarithms, it must be
so expressed for the generalized structure theorem to apply.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Rothstein 76b]{Ro76b} Rothstein, Michael; Caviness, B.F.
``A structure theorem for exponential and primitive functions''
SIAM J. Computing Vol 8 No 3 (1979)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Ro76b.pdf REF:00104
+\bibitem[Sit 92]{REFSit92} Sit, William
+``An Algorithm for Parametric Linear Systems''
+J. Sym. Comp., April 1992
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this paper a new theorem is proved that generalizes a result of
Risch. The new theorem gives all the possible algebraic relationships
among functions that can be built up from the rational functions by
algebraic operations, by taking exponentials, and by integration. The
functions so generated are called exponential and primitive functions.
From the theorem an algorithm for determining algebraic dependence
among a given set of exponential and primitive functions is derived.
The algorithm is then applied to a problem in computer algebra.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{Roth77,
 author = "Rothstein, Michael",
 title = "A new algorithm for the integration of exponential and logarithmic functions",
 journal = "Proceedings of the 1977 MACSYMA Users Conference",
 year = "1977",
 pages = "263274",
 publisher = "NASA Pub CP2012"
}
+\begin{chunk}{ignore}
+\bibitem[Smith 67]{Smi67} Smith B T.
+``ZERPOL: A Zero Finding Algorithm for Polynomials Using Laguerre's Method''
+Technical Report. Department of Computer Science, University of Toronto,
+Canada. (1967)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Seidenberg 58]{Sei58} Seidenberg, Abraham
``Abstract differential algebra and the analytic case''
Proc. Amer. Math. Soc. Vol 9 pp159164 (1958)
+\bibitem[Smith 85]{Smi85} Smith G D.
+``Numerical Solution of Partial Differential Equations: Finite Difference
+Methods''
+Oxford University Press (3rd Edition). (1985)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Seidenberg 69]{Sei69} Seidenberg, Abraham
``Abstract differential algebra and the analytic case. II''
Proc. Amer. Math. Soc. Vol 23 pp689691 (1969)
+\bibitem[Sobol 74]{Sob74} Sobol I M.
+``The Monte Carlo Method''
+The University of Chicago Press. 1974
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Singer 85]{Sing85} Singer, M.F.; Saunders, B.D.; Caviness, B.F.
``An extension of Liouville's theorem on integration in finite terms''
SIAM J. of Comp. Vol 14 pp965990 (1985)
\verbwww4.ncsu.edu/~singer/papers/singer_saunders_caviness.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Sing85.pdf
+\bibitem[Steele 90]{Ste90} Steele, Guy L.
+``Common Lisp The Language''
+Second Edition ISBN 1555580416 Digital Press (1990)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In Part 1 of this paper, we give an extension of Liouville's Theorem
and give a number of examples which show that integration with special
functions involves some phenomena that do not occur in integration
with the elementary functions alone. Our main result generalizes
Liouville's Theorem by allowing, in addition to the elementary
functions, special functions such as the error function, Fresnel
integrals and the logarithmic integral (but not the dilogarithm or
exponential integral) to appear in the integral of an elementary
function. The basic conclusion is that these functions, if they
appear, appear linearly. We give an algorithm which decides if an
elementary function, built up using only exponential functions and
rational operations has an integral which can be expressed in terms of
elementary functions and error functions.
\end{adjustwidth}
+\begin{chunk}{axiom.bib}
+@misc{Stic93,
+ author = "Stichtenoth, H.",
+ title = "Algebraic function fields and codes",
+ publisher = "SpringerVerlag",
+ year = "1993"
+}
+
+\end{chunk}
\begin{chunk}{ignore}
\bibitem[Slagle 61]{Slag61} Slagle, J.
``A heuristic program that solves symbolic integration problems in freshman calculus''
Ph.D Diss. MIT, May 1961; also Computers and Thought, Feigenbaum and Feldman.
% REF:00014
+\bibitem[Stinson 90]{Stin90} Stinson, D.R.
+``Some observations on parallel Algorithms for fast exponentiation
+in $GF(2^n)$''
+Siam J. Comp., Vol.19, No.4, pp.711717, August 1990
+%\verbaxiomdeveloper.org/axiomwebsite/Stin90.pdf
+ abstract = "
+ A normal basis represention in $GF(2^n)$ allows squaring to be
+ accomplished by a cyclic shift. Algorithms for multiplication in
+ $GF(2^n)$ using a normal basis have been studied by several
+ researchers. In this paper, algorithms for performing exponentiation
+ in $GF(2^n)$ using a normal basis, and how they can be speeded up by
+ using parallelization, are investigated."
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Terelius 09]{Tere09} Terelius, Bjorn
``Symbolic Integration''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Tere09.pdf
+\bibitem[Stroud 66]{SS66} Stroud A H.; Secrest D.
+``Gaussian Quadrature Formulas''
+PrenticeHall. (1966)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Symbolic integration is the problem of expressing an indefinite integral
$\int{f}$ of a given function $f$ as a finite combination $g$ of elementary
functions, or more generally, to determine whether a certain class of
functions contains an element $g$ such that $g^\prime = f$.

In the first part of this thesis, we compare different algorithms for
symbolic integration. Specifically, we review the integration rules
taught in calculus courses and how they can be used systematically to
create a reasonable, but somewhat limited, integration method. Then we
present the differential algebra required to prove the transcendental
cases of Risch's algorithm. Risch's algorithm decides if the integral
of an elementary function is elementary and if so computes it. The
presentation is mostly selfcontained and, we hope, simpler than
previous descriptions of the algorithm. Finally, we describe
RischNorman's algorithm which, although it is not a decision
procedure, works well in practice and is considerably simpler than the
full Risch algorithm.
+\begin{chunk}{ignore}
+\bibitem[Stroud 71]{Str71} Stroud A H.
+``Approximate Calculation of Multiple Integrals''
+PrenticeHall 1971
In the second part of this thesis, we briefly discuss an
implementation of a computer algebra system and some of the
experiences it has given us. We also demonstrate an implementation of
the rulebased approach and how it can be used, not only to compute
integrals, but also to generate readable derivations of the results.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{axiom.bib}
@article{Trag76,
 author = "Trager, Barry",
 title = "Algebraic factoring and rational function integration",
 journal = "Proceedings of SYMSAC'76",
 year = "1976",
 pages = "219226",
 paper = "Trag76.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Swarztrauber 79]{SS79} Swarztrauber P N.; Sweet R A.
+``Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial
+Differential Equations''
+ACM Trans. Math. Softw. 5 352364. (1979)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This paper presents a new, simple, and efficient algorithm for
factoring polynomials in several variables over an algebraic number
field. The algorithm is then used interatively to construct the
splitting field of a polynomial over the integers. Finally the
factorization and splitting field algorithms are applied to the
problem of determining the transcendental part of the integral of a
rational function. In particular, a constructive procedure is given
for finding a least degree extension field in which the integral can
be expressed.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Trager 76a]{Tr76a} Trager, Barry Marshall
``Algorithms for Manipulating Algebraic Functions''
MIT Master's Thesis.
\verbwww.dm.unipi.it/pages/gianni/public_html/AlgComp/fattorizzazioneEA.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Tr76a.pdf REF:00050
+\bibitem[Swarztrauber 84]{SS84} Swarztrauber P N.
+``Fast Poisson Solvers''
+Studies in Numerical Analysis. (ed G H Golub)
+Mathematical Association of America. (1984)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Given a base field $k$, of characteristic zero, with effective
procedures for performing arithmetic and factoring polynomials, this
thesis presents algorithms for extending those capabilities to
elements of a finite algebraic symbolic manipulation system. An
algebraic factorization algorithm along with a constructive version of
the primitive element theorem is used to construct splitting fields of
polynomials. These fields provide a context in which we can operate
symbolically with all the roots of a set of polynomials. One
application for this capability is rational function integrations.
Previously presented symbolic algorithms concentrated on finding the
rational part and were only able to compute the complete
integral in special cases. This thesis presents an algorithm for
finding an algebraic extension field of least degreee in which the
integral can be expressed, and then constructs the integral in that
field. The problem of algebraic function integration is also
examined, and a highly efficient procedure is presented for generating
the algebraic part of integrals whose function fields are defined by a
single radical extension of the rational functions.
\end{adjustwidth}
+\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@phdthesis{Trag84,
 author = "Trager, Barry",
 title = "On the integration of algebraic functions",
 school = "MIT",
 year = "1984",
 url = "http://www.dm.unipi.it/pages/gianni/public_html/AlgComp/thesis.pdf",
 paper = "Trag76.pdf"
+@book{Tait1890,
+ author = "Tait, P.G.",
+ title = "An Elementary Treatise on Quaternions",
+ publisher = "C.J. Clay and Sons, Cambridge University Press Warehouse,
+ Ave Maria Lane",
+ year = "1890"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We show how the ``rational'' approach for integrating algebraic
functions can be extended to handle elementary functions. The
resulting algorithm is a practical decision procedure for determining
whether a given elementary function has an elementary antiderivative,
and for computing it if it exists.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[W\"urfl 07]{Wurf07} W\"urfl, Andreas
``Basic Concepts of Differential Algebra''
\verbwww14.in.tum.de/konferenzen/Jass07/courses/1/Wuerfl/wuerfl_paper.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Wurf07.pdf
+\bibitem[Taivalsaari 96]{Tai96} Taivalsaari, Antero
+``On the Notion of Inheritance''
+ACM Computing Surveys, Vol 28 No 3 Sept 1996 pp438479
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Modern computer algebra systems symbolically integrate a vast variety
of functions. To reveal the underlying structure it is necessary to
understand infinite integration not only as an analytical problem but
as an algebraic one. Introducing the differential field of elementary
functions we sketch the mathematical tools like Liouville's Principle
used in modern algorithms. We present Hermite's method for integration
of rational functions as well as the Rothstein/Trager method for
rational and for elementary functions. Further applications of the
mentioned algorithms in the field of ODE's conclude this paper.
\end{adjustwidth}

\subsection{Partial Fraction Decomposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Angell]{Angell} Angell, Tom
``Guidelines for Partial Fraction Decomposition''
\verbwww.math.udel.edu/~angell/partfrac_I.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Angell.pdf
+\bibitem[Temme 87]{Tem87} Temme N M.
+``On the Computation of the Incomplete Gamma Functions for Large Values of
+the Parameters''
+Algorithms for Approximation. (ed J C Mason and M G Cox)
+Oxford University Press. (1987)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Laval 08]{Lava08} Laval, Philippe B.
``Partial Fractions Decomposition''
\verbwww.math.wisc.edu/~park/Fall2011/integration/Partial%20Fraction.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Lava08.pdf
+\bibitem[Temperton 83a]{Tem83a} Temperton C.
+``Selfsorting Mixedradix Fast Fourier Transforms''
+J. Comput. Phys. 52 123. (1983)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Mudd 14]{Mudd14} Harvey Mudd College
``Partial Fractions''
\verbwww.math.hmc.edu/calculus/tutorials/partial_fractions/partial_fractions.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Mudd14.pdf
+\bibitem[Temperton 83b]{Tem83b} Temperton C.
+``Fast MixedRadix Real Fourier Transforms''
+J. Comput. Phys. 52 340350. (1983)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rajasekaran 14]{Raja14} Rajasekaran, Raja
``Partial Fraction Expansion''
\verbwww.utdallas.edu/~raja1/EE4361%20Spring%2014/Lecture%20Notes/
\verbPartial%20Fractions.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Raja14.pdf
+\begin{chunk}{axiom.bib}
+@article{Thur94,
+ author = "Thurston, William P.",
+ title = "On Proof and Progress in Mathematics",
+ journal = "Bulletin AMS",
+ volume = "30",
+ number = "2",
+ month = "April",
+ year = "1994",
+ url = "http://www.ams.org/journals/bull/19943002/S027309791994005026/S027309791994005026.pdf",
+ paper = "Thur94.pdf"
+}
\end{chunk}
+\subsection{U} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Wootton 14]{Woot14} Wootton, Aaron
``Integration of Rational Functions by Partial Fractions''
\verbfaculty.up.edu/wootton/calc2/section7.4.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Woot14.pdf
+\bibitem[Unknown 61]{Unk61} Unknown
+``Chebyshevseries''
+Modern Computing Methods
+Chapter 8. NPL Notes on Applied Science (2nd Edition). 16 HMSO. 1961
\end{chunk}
\subsection{Ore Rings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This is used as a reference for the LeftOreRing category, in particular,
the least left common multiple (lcmCoef) function.
+\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Abramov 97]{Abra97} Abramov, Sergei A.; van Hoeij, Mark
``A method for the Integration of Solutions of Ore Equations''
Proc ISSAC 97 pp172175 (1997)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Abra97.pdf
+\bibitem[Van Dooren 76]{vDDR76} Van Dooren P.; De Ridder L.
+``An Adaptive Algorithm for Numerical Integration over an Ndimensional
+Cube''
+J. Comput. Appl. Math. 2 207217. (1976)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We introduce the notion of the adjoint Ore ring and give a definition
of adjoint polynomial, operator and equation. We apply this for
integrating solutions of Ore equations.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Delenclos 06]{DL06} Delenclos, Jonathon; Leroy, Andr\'e
``Noncommutative Symmetric functions and $W$polynomials''
\verbarxiv.org/pdf/math/0606614.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/DL06.pdf
+\bibitem[van Hoeij 94]{REFvH94} van Hoeij, M.
+``An algorithm for computing an integral
+basis in an algebraic function field''
+{\sl J. Symbolic Computation}
+18(4):353364, October 1994
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Let $K$, $S$, $D$ be a division ring an endomorphism and a
$S$derivation of $K$, respectively. In this setting we introduce
generalized noncommutative symmetric functions and obtain Vi\'ete
formula and decompositions of different operators. $W$polynomials
show up naturally, their connetions with $P$independency. Vandermonde
and Wronskian matrices are briefly studied. The different linear
factorizations of $W$polynomials are analysed. Connections between
the existence of LLCM (least left common multiples) of monic linear
polynomials with coefficients in a ring and the left duo property are
established at the end of the paper.
\end{adjustwidth}

\begin{chunk}{ignore}
\bibitem[Abramov 05]{Abra05} Abramov, S.A.; Le, H.Q.; Li, Z.
``Univariate Ore Polynomial Rings in Computer Algebra''
\verbwww.mmrc.iss.ac.cn/~zmli/papers/oretools.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Abra05.pdf
+\bibitem[Van Loan 92]{Van92} Van Loan, C.
+``Computational Frameworks for the Fast Fourier Transform''
+SIAM Philadelphia. (1992)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present some algorithms related to rings of Ore polynomials (or,
briefly, Ore rings) and describe a computer algebra library for basic
operations in an arbitrary Ore ring. The library can be used as a
basis for various algorithms in Ore rings, in particular, in
differential, shift, and $q$shift rings.
\end{adjustwidth}

\subsection{Number Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Shoup 08]{Sho08} Shoup, Victor
``A Computational Introduction to Number Theory''
\verbshoup.net/ntb/ntbv2.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Sho08.pdf
+\bibitem[Wait 85]{WM85} Wait R.; Mitchell A R.
+``Finite Element Analysis and Application''
+Wiley. (1985)
\end{chunk}
\subsection{Polynomial Factorization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Branch Cuts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{axiom.bib}
@article{Beau03,
 author = "Beaumont, James and Bradford, Russell and Davenport, James H.",
 title = "Better simplification of elementary functions through power series",
 journal = "2003 International Symposium on Symbolic and Algebraic Computation",
 series = "ISSAC'03",
 year = "2003",
 month = "August",
 paper = "Beau03.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Wang 92]{Wang92} Wang, D.M.
+``An implementation of the characteristic set method in Maple''
+Proc. DISCO'92 Bath, England
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In [5], we introduced an algorithm for deciding whether a proposed
simplification of elementary functions was correct in the presence of
branch cuts. This algorithm used multivalued function simplification
followed by verification that the branches were consistent.

In [14] an algorithm was presented for zerotesting functions defined
by ordinary differential equations, in terms of their power series.

The purpose of the current paper is to investigate merging the two
techniques. In particular, we will show an explicit reduction to the
constant problem [16].
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Ward 75]{War75} Ward, R C.
+``The Combination Shift QZ Algorithm''
+SIAM J. Numer. Anal. 12 835853. 1975
\begin{chunk}{axiom.bib}
@article{Beau07,
 author = "Beaumont, James C. and Bradford, Russell J. and Davenport, James H. and Phisanbut, Nalina",
 title = "Testing elementary function identities using CAD",
 journal = "Applicable Algebra in Engineering, Communication and Computing",
 year = "2007",
 volume = "18",
 number = "6",
 issn = "09381279",
 publisher = "SpringerVerlag",
 pages = "513543",
 paper = "Beau07.pdf"
}

\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
One of the problems with manipulating function identities in computer
algebra systems is that they often involve functions which are
multivalued, whilst most users tend to work with singlevalued
functions. The problem is that many wellknown identities may no
longer be true everywhere in the complex plane when working with their
singlevalued counterparts. Conversely, we cannot ignore them, since
in particular contexts they may be valid. We investigate the
practicality of a method to verify such identities by means of an
experiment; this is based on a set of test examples which one might
realistically meet in practice. Essentially, the method works as
follows. We decompose the complex plane via means of cylindrical
algebraic decomposition into regions with respect to the branch cuts
of the functions. We then test the identity numerically at a sample
point in the region. The latter step is facilitated by the notion of
the {\sl adherence} of a branch cut, which was previously introduced
by the authors. In addition to presenting the results of the
experiment, we explain how adherence relates to the proposal of
{\sl signed zeros} by W. Kahan, and develop this idea further in order to
allow us to cover previously untreatable cases. Finally, we discuss
other ways to improve upon our general methodology as well as topics
for future research.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{Brad02,
 author="Bradford, Russell and Corless, RobertM. and Davenport, JamesH. and Jeffrey, DavidJ. and Watt, StephenM.",
 title="Reasoning about the Elementary Functions of Complex Analysis",
 journal="Annals of Mathematics and Artificial Intelligence",
 year="2002",
 issn="10122443",
 volume="36",
 number="3",
 doi="10.1023/A:1016007415899",
 url="http://dx.doi.org/10.1023/A%3A1016007415899",
 publisher="Kluwer Academic Publishers",
 keywords="elementary functions; branch cuts; complex identities",
 pages="303318",
 paper = "Brad02.pdf"
+@misc{Watt03,
+ author = "Watt, Stephen",
+ title = "Aldor",
+ url = "http://www.aldor.org",
+ year = "2003"
}
\end{chunk}

\begin{adjustwidth}{2.5em}{0pt}
There are many problems with the simplification of elementary
functions, particularly over the complex plane, though not
exclusively. 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 complex
variable analysis.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@inproceedings{Chyz11,
 author = "Chyzak, Fr\'ed\'eric and Davenport, James H. and Koutschan, Christoph and Salvy, Bruno",
 title = "On Kahan's Rules for Determining Branch Cuts",
 booktitle = "Proc. 13th Int. Symp. on Symbolic and Numeric Algorithms for Scientific Computing",
 year = "2011",
 isbn = "9781467302074",
 location = "Timisoara",
 pages = "4751",
 doi = "10.1109/SYNASC.2011.51",
 acmid = "258794",
 publisher = "IEEE",
 paper = "Chyz11.pdf"
+
+\begin{chunk}{axiom.bib}
+@misc{Weil71,
+ author = "Weil, Andr\'{e}",
+ title = "Courbes alg\'{e}briques et vari\'{e}t\'{e}s Abeliennes",
+ year = "1971"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In computer algebra there are different ways of approaching the
mathematical concept of functions, one of which is by defining them as
solutions of differential equations. We compare different such
appraoches and discuss the occurring problems. The main focus is on
the question of determining possible branch cuts. We explore the
extent to which the treatment of branch cuts can be rendered (more)
algorithmic, by adapting Kahan's rules to the differential equation
setting.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Weisstein]{Wein} Weisstein, Eric W.
+``Hypergeometric Function''
+MathWorld  A Wolfram Web Resource
+\verbmathworld.wolfram.com/HypergeometricFunction.html
+
+\end{chunk}
\begin{chunk}{axiom.bib}
@article{Dave10,
 author = "Davenport, James",
 title = {The Challenges of Multivalued "Functions"},
 journal = "Lecture Notes in Computer Science",
 volume = "6167",
 year = "2010",
 pages = "112",
 paper = "Dave10.pdf"
+@misc{Weit03,
+ author = "Weitz, E.",
+ title = "CLWHO Yet another Lisp markup language",
+ year = "2003",
+ url = "http://www.weitz.de/clwho/"
}

\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Although, formally, mathematics is clear that a function is a
singlevalued object, mathematical practice is looser, particularly
with nth roots and various inverse functions. In this paper, we point
out some of the looseness, and ask what the implications are, both for
Artificial Intelligence and Symbolic Computation, of these practices.
In doing so, we look at the steps necessary to convert existing tests
into
\begin{itemize}
\item (a) rigorous statements
\item (b) rigorously proved statements
\end{itemize}
In particular we ask whether there might be a constant ``de Bruij factor''
[18] as we make these texts more formal, and conclude that the answer
depends greatly on the interpretation being placed on the symbols.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{axiom.bib}
@article{Dave12,
 author = "Davenport, James H. and Bradford, Russell and England, Matthew and Wilson, David",
 title = "Program Verification in the presence of complex numbers, functions with branch cuts etc",
 journal = "14th Int. Symp. on Symbolic and Numeric Algorithms for Scientific Computing",
 year = "2012",
 series = "SYNASC'12",
 pages = "8388",
 publisher = "IEEE",
 paper = "Dave12.pdf"
+@misc{Weit06,
+ author = "Weitz, E.",
+ title = "HUNCHENTOOT  The Common Lisp web server formerly known as TBNL",
+ year = "2006",
+ url = "http://www.weitz.de/hunchentoot"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In considering the reliability of numerical programs, it is normal to
``limit our study to the semantics dealing with numerical precision''.
On the other hand, there is a great deal of work on the reliability of
programs that essentially ignores the numerics. The thesis of this
paper is that there is a class of problems that fall between the two,
which could be described as ``does the lowlevel arithmetic implement
the highlevel mathematics''. Many of these problems arise because
mathematics, particularly the mathematics of the complex numbers, is
more difficult than expected; for example the complex function log is
not continuous, writing down a program to compute an inverse function
is more complicated than just solving an equation, and many algebraic
simplification rules are not universally valid.
+\begin{chunk}{ignore}
+\bibitem[Wesseling 82a]{Wes82a} Wesseling, P.
+``MGD1  A Robust and Efficient Multigrid Method''
+Multigrid Methods. Lecture Notes in Mathematics. 960
+SpringerVerlag. 614630. (1982)
The good news is that these problems are theoretically capable of
being solved, and are practically close to being solved, but not yet
solved, in several realworld examples. However, there is still a long
way to go before implementations match the theoretical possibilities.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{axiom.bib}
@article{Jeff04,
 author = "Jeffrey, D. J. and Norman, A. C.",
 title = "Not Seeing the Roots for the Branches: Multivalued Functions in Computer Algebra",
 journal = "SIGSAM Bull.",
 issue_date = "September 2004",
 volume = "38",
 number = "3",
 month = "September",
 year = "2004",
 issn = "01635824",
 pages = "5766",
 numpages = "10",
 url = "http://doi.acm.org/10.1145/1040034.1040036",
 doi = "10.1145/1040034.1040036",
 acmid = "1040036",
 publisher = "ACM",
 address = "New York, NY, USA",
 paper = "Jeff04.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Wesseling 82b]{Wes82b} Wesseling, P.
+``Theoretical Aspects of a Multigrid Method''
+SIAM J. Sci. Statist. Comput. 3 387407. (1982)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We discuss the multiple definitions of multivalued functions and their
suitability for computer algebra systems. We focus the discussion by
taking one specific problem and considering how it is solved using
different definitions. Our example problem is the classical one of
calculating the roots of a cubic polynomial from the Cardano formulae,
which contains fractional powers. We show that some definitions of
these functions result in formulae that are correct only in the sense
that they give candidates for solutions; these candidates must then be
tested. Formulae that are based on singlevalued functions, in
contract, are efficient and direct.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Wicks 89]{Wic89} Wicks, Mark; Carlisle, David, Rahtz, Sebastian
+``dvipdfm.def''
+\verbweb.mit.edu/texsrc/source/latex/graphics/dvipdfm.def
\begin{chunk}{axiom.bib}
@inproceedings{Kaha86,
 author = "Kahan, W.",
 title = "Branch cuts for complex elementary functions",
 booktitle = "The State of the Art in Numerical Analysis",
 year = "1986",
 month = "April",
 editor = "Powell, M.J.D and Iserles, A.",
 publisher = "Oxford University Press"
}
+\end{chunk}
\end{chunk}
+\begin{chunk}{ignore}
+\bibitem[Wiki 3]{Wiki3}.
+``Givens Rotations''
+\verben.wikipedia.org/wiki/Givens_rotation
+
+\end{chunk}
\begin{chunk}{axiom.bib}
@article{Rich96,
 author = "Rich, Albert D. and Jeffrey, David J.",
 title = "Function Evaluation on Branch Cuts",
 journal = "SIGSAM Bull.",
 issue_date = "June 1996",
 volume = "30",
 number = "2",
 month = "June",
 year = "1996",
 issn = "01635824",
 pages = "2527",
 numpages = "3",
 url = "http://doi.acm.org/10.1145/235699.235704",
 doi = "10.1145/235699.235704",
 acmid = "235704",
 publisher = "ACM",
 address = "New York, NY, USA"
+@misc{Wiki14a,
+ author = "ProofWiki",
+ title = "Euclidean Algorithm",
+ url = "http://proofwiki.org/wiki/Euclidean_Algorithm"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Once it is decided that a CAS will evaluate multivalued functions on
their principal branches, questions arise concerning the branch
definitions. The first questions concern the standardization of the
positions of the branch cuts. These questions have largely been
resolved between the various algebra systems and the numerical
libraries, although not completely. In contrast to the computer
systems, many mathematical textbooks are much further behind: for
example, many popular textbooks still specify that the argument of a
complex number lies between 0 and $2\pi$. We do not intend to discuss
these first questions here, however. Once the positions of the branch
cuts have been fixed, a second set of questions arises concerning the
evaluation of functions on their branch cuts.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{Patt96,
 author = "Patton, Charles M.",
 title = "A Representation of Branchcut Information",
 journal = "SIGSAM Bull.",
 issue_date = "June 1996",
 volume = "30",
 number = "2",
 month = "June",
 year = "1996",
 issn = "01635824",
 pages = "2124",
 numpages = "4",
 url = "http://doi.acm.org/10.1145/235699.235703",
 doi = "10.1145/235699.235703",
 acmid = "235703",
 publisher = "ACM",
 address = "New York, NY, USA",
 paper = "Patt96.pdf"
+@misc{Wiki14b,
+ author = "ProofWiki",
+ title = "Division Theorem",
+ url = "http://proofwiki.org/wiki/Division_Theorem"
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
Handling (possibly) multivalued functions is a problem in all current
computer algebra systems. The problem is not an issue of technology.
Its solution, however, is tied to a uniform handling of the issues by
the mathematics community.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@article{Squi91,
 author = "Squire, Jon S.",
 title = "Rationale for the Proposed Standard for a Generic Package of Complex Elementary Functions",
 journal = "Ada Lett.",
 issue_date = "Fall 1991",
 volume = "XI",
 number = "7",
 month = "September",
 year = "1991",
 issn = "10943641",
 pages = "166179",
 numpages = "14",
 url = "http://doi.acm.org/10.1145/123533.123545",
 doi = "10.1145/123533.123545",
 acmid = "123545",
 publisher = "ACM",
 address = "New York, NY, USA",
 paper = "Squi91.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Williamson 85]{Wil85} Williamson, S.G.
+``Combinatorics for Computer Science''
+Computer Science Press, 1985.
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This document provides the background on decisions that were made
during the development of the specification for Generic Complex
Elementary fuctions. It also rovides some information that was used to
develop error bounds, range, domain and definitions of complex
elementary functions.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 71]{WR71} Wilkinson J H.; Reinsch C.
+``Handbook for Automatic Computation II, Linear Algebra''
+SpringerVerlag. 1971
+
+\end{chunk}
\begin{chunk}{axiom.bib}
@article{Squi91a,
 editor = "Squire, Jon S.",
 title = "Proposed Standard for a Generic Package of Complex Elementary Functions",
 journal = "Ada Lett.",
 issue_date = "Fall 1991",
 volume = "XI",
 number = "7",
 month = "September",
 year = "1991",
 issn = "10943641",
 pages = "140165",
 numpages = "26",
 url = "http://doi.acm.org/10.1145/123533.123544",
 doi = "10.1145/123533.123544",
 acmid = "123544",
 publisher = "ACM",
 address = "New York, NY, USA"
}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 63]{Wil63} Wilkinson J H.
+``Rounding Errors in Algebraic Processes''
+ Chapter 2. HMSO. (1963)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
This document defines the specification of a generic package of
complex elementary functions called Generic Complex Elementary
Functions. It does not provide the body of the package.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 65]{Wil65} Wilkinson J H.
+``The Algebraic Eigenvalue Problem''
+ Oxford University Press. (1965)
\subsection{Squarefree Decomposition } %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{axiom.bib}
@article{Bern97,
 author = "Bernardin, Laurent",
 title = "On squarefree factorization of multivariate polynomials over a finite field",
 journal = "Theoretical Computer Science",
 volume = "187",
 number = "12",
 year = "1997",
 month = "November",
 pages = "105116",
 keywords = "axiomref",
 paper = "Bern97.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 78]{Wil78} Wilkinson J H.
+``Singular Value Decomposition  Basic Aspects''
+Numerical Software  Needs and Availability.
+(ed D A H Jacobs) Academic Press. (1978)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In this paper we present a new deterministic algorithm for computing
the squarefree decomposition of multivariate polynomials with
coefficients from a finite field.
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 79]{Wil79} Wilkinson J H.
+``Kronecker's Canonical Form and the QZ Algorithm''
+Linear Algebra and Appl. 28 285303. 1979
Our algorithm is based on Yun's squarefree factorization algorithm
for characteristic 0. The new algorithm is more efficient than
existing, deterministic algorithms based on Musser's squarefree
algorithm
+\end{chunk}
We will show that the modular approach presented by Yun has no
significant performance advantage over our algorithm. The new
algorithm is also simpler to implement and it can rely on any existing
GCD algorithm without having to worry about choosing "good" evaluation
points.
+\begin{chunk}{ignore}
+\bibitem[Wisbauer 91]{Wis91} Wisbauer, R.
+``Bimodule Structure of Algebra''
+Lecture Notes Univ. Duesseldorf 1991
To demonstrate this, we present some timings using implementations in
Maple (Char et al. 1991), where the new algorithm is used for Release
4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system
known to the author to use and implementation of Yun's modular
algorithm mentioned above.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{axiom.bib}
@article{Chez07,
 author = "Ch\'eze, Guillaume and Lecerf, Gr\'egoire",
 title = "Lifting and recombination techniques for absolute factorization",
 journal = "Journal of Complexity",
 volume = "23",
 number = "3",
 year = "2007",
 month = "June",
 pages = "380420",
 paper = "Chez07.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[WoerzBusekros 80]{Woe80} WoerzBusekros, A.
+``Algebra in Genetics''
+Lectures Notes in Biomathematics 36, SpringerVerlag, Heidelberg, 1980
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
In the vein of recent algorithmic advances in polynomial factorization
based on lifting and recombination techniques, we present new faster
algorithms for computing the absolute factorization of a bivariate
polynomial. The running time of our probabilistic algorithm is less
than quadratic in the dense size of the polynomial to be factored.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Wolberg 67]{Wol67} Wolberg J R.
+``Prediction Analysis''
+Van Nostrand. (1967)
\begin{chunk}{axiom.bib}
@article{Lece07,
 author = "Lecerf, Gr\'egoire",
 title = "Improved dense multivariate polynomial factorization algorithms",
 journal = "Journal of Symbolic Computation",
 volume = "42",
 number = "4",
 year = "2007",
 month = "April",
 pages = "477494",
 paper = "Lece07.pdf"
}
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Wolfram 09]{Wo09} Wolfram Research
+\verbmathworld.wolfram.com/Quaternion.html
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We present new deterministic and probabilistic algorithms that reduce
the factorization of dense polynomials from several variables to one
variable. The deterministic algorithm runs in subquadratic time in
the dense size of the input polynomial, and the probabilistic
algorithm is softly optimal when the number of variables is at least
three. We also investigate the reduction from several to two variables
and improve the quantitative versions of Bertini's irreducibility theorem.
\end{adjustwidth}
+\begin{chunk}{ignore}
+\bibitem[Wu 87]{WU87} Wu, W.T.
+``A Zero Structure Theorem for polynomial equations solving''
+MM Research Preprints, 1987
\begin{chunk}{axiom.bib}
@article{Wang77,
 author = "Wang, Paul S.",
 title = "An efficient squarefree decomposition algorithm",
 journal = "ACM SIGSAM Bulletin",
 volume = "11",
 number = "2",
 year = "1977",
 month = "May",
 pages = "46",
 paper = "Wang77.pdf"
}
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Wynn 56]{Wynn56} Wynn P.
+``On a Device for Computing the $e_m(S_n )$ Transformation''
+Math. Tables Aids Comput. 10 9196. (1956)
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
The concept of polynomial squarefree decomposition is an important one
in algebraic computation. The squarefree decomposition process has
many uses in computer symbolic computation. A recent survey by D. Yun
[3] describes many useful algorithms for this purpose. All of these
methods depend on computing the greated common divisor (gcd) of the
polynomial to be decomposed and its first derivative (with repect to
some variable). In the multivariate case, this gcd computation is
nontrivial and dominates the cost for the squarefree decompostion.
\end{adjustwidth}
+\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@article{Wang79,
 author = "Wang, Paul S. and Trager, Barry M.",
 title = "New Algorithms for Polynomial SquareFree Decomposition over the Integers",
 journal = "SIAM Journal on Computing",
 volume = "8",
 number = "3",
 year = "1979",
 publisher = "Society for Industrial and Applied Mathematics",
 issn = "00975397",
 paper = "Wang79.pdf"
}
+\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{chunk}
+\begin{chunk}{ignore}
+\bibitem[Zakrajsek 02]{Zak02} Zakrajsek, Helena
+``Applications of Hermite transform in computer algebra''
+\verbwww.imfm.si/preprinti/PDF/00835.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Zak02.pdf
+ abstract = "
+ let $L$ be a linear differential operator with polynomial
+ coefficients. We show that there is an isomorphism of differential
+ operators ${\bf D_\alpha}$ and an integral transform ${\bf H_\alpha}$
+ (called the Hermite transform) on functions for which $({\bf
+ D_\alpha}{\bf L})f(x)=0$ implies ${\bf L}{\bf H_alpha}(f)(x)=0$. We
+ present an algorithm that computes the Hermite transform of a rational
+ function and use it to find $n+1$ linearly independent solutions of
+ ${\bf L}y=0$ when $({\bf D_\alpha}{\bf L})f(x)=0$ has a rational
+ solution with $n$ distinct finite poles."
\begin{adjustwidth}{2.5em}{0pt}
Previously known algorithms for polynomial squarefree decomposition
rely on greatest common divisor (gcd) computations over the same
coefficient domain where the decomposition is to be performed. In
particular, gcd of the given polynomial and its first derivative (with
respect to some variable) is obtained to begin with. Application of
modular homomorphism and $p$adic construction (multivariate case) or
the Chinese remainder algorithm (univariate case) results in new
squarefree decomposition algorithms which, generally speaking, take
less time than a single gcd between the given polynomial and its first
derivative. The key idea is to obtain one or several ``correct''
homomorphic images of the desired squarefree decomposition
first. This provides information as to how many different squarefree
factors there are, their multiplicities and their homomorphic
images. Since the multiplicities are known, only the squarefree
factors need to be constructed. Thus, these new algorithms are
relatively insensitive to the multiplicities of the squarefree factors.
\end{adjustwidth}
+\end{chunk}
\begin{chunk}{axiom.bib}
@inproceedings{Yun76,
 author = "Yun, D.Y.Y",
 title = "On squarefree decomposition algorithms",
 booktitle = "Proceedings of SYMSAC'76",
 year = "1976",
 keywords = "survey",
 pages = "2635"
+@misc{Zdan14,
+ author = "Zdancewic, Steve and Martin, Milo M.K.",
+ title = "Vellvm: Verifying the LLVM",
+ url = "http://www.cis.upenn.edu/~stevez/vellvm"
}
\end{chunk}
+\begin{chunk}{ignore}
+\bibitem[Zhi 97]{Zhi97} Zhi, Lihong
+``Optimal Algorithm for Algebraic Factoring''
+\verbwww.mmrc.iss.ac.cn/~lzhi/Publications/zopfac.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Zhi97.pdf
+ abstract = "
+ This paper presents an optimized method for factoring multivariate
+ polynomials over algebraic extension fields which defined by an
+ irreducible ascending set. The basic idea is to convert multivariate
+ polynomials to univariate polynomials and algebraic extensions fields
+ to algebraic number fields by suitable integer substitutions, then
+ factorize the univariate polynomials over the algebraic number fields.
+ Finally, construct multivariate factors of the original polynomial by
+ Hensel lemma and TRUEFACTOR test. Some examples with timing are
+ included."
+
+\end{chunk}
\subsection{To Be Classified} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@PhdThesis{Kalt82,
author = "Kaltofen, E.",
 title = "On the complexity of factoring polynomials with integer coefficients",
+ title = "On the complexity of factoring polynomials with integer
+ coefficients",
school = "RPI",
address = "Troy, N. Y.",
year = "1982",
@@ 10712,7 +10341,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt82a,
author = "Kaltofen, E.",
 title = "A polynomialtime reduction from bivariate to univariate integral polynomial factorization",
+ title = "A polynomialtime reduction from bivariate to univariate
+ integral polynomial factorization",
booktitle = "Proc. 23rd Annual Symp. Foundations of Comp. Sci.",
year = "1982",
pages = "5764",
@@ 10772,11 +10402,13 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt84a,
author = "Kaltofen, E. and Yui, N.",
 title = "Explicit construction of the {Hilbert} class field of imaginary quadratic fields with class number 7 and 11",
+ title = "Explicit construction of the {Hilbert} class field of imaginary
+ quadratic fields with class number 7 and 11",
booktitle = "Proc. EUROSAM '84",
pages = "310320",
crossref = "EUROSAM84",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/84/KaYui84_eurosam.ps.gz",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/84/KaYui84_eurosam.ps.gz",
paper = "Kalt84a.ps"
}
@@ 10789,7 +10421,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
institution = "RPI",
address = "Dept. Comput. Sci., Troy, New York",
year = "1984",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_integration.pdf",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_integration.pdf",
paper = "Kalt84b.pdf"
}
@@ 10802,7 +10435,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
booktitle = "Proc. EUROSAM '84",
pages = "275284",
crossref = "EUROSAM84",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_infcontr.ps.gz",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_infcontr.ps.gz",
paper = "Kalt85.ps"
}
@@ 10826,7 +10460,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt85b,
author = "Kaltofen, E.",
 title = "Computing with polynomials given by straightline programs {II}; sparse factorization",
+ title = "Computing with polynomials given by straightline programs {II};
+ sparse factorization",
booktitle = "Proc. 26th Annual Symp. Foundations of Comp. Sci.",
year = "1985",
pages = "451458",
@@ 10867,7 +10502,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Kalt85e,
author = "Kaltofen, E.",
 title = "Polynomialtime reductions from multivariate to bi and univariate integral polynomial factorization",
+ title = "Polynomialtime reductions from multivariate to bi and univariate
+ integral polynomial factorization",
journal = "{SIAM} J. Comput.",
year = "1985",
volume = "14",
@@ 10887,7 +10523,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
year = "1985",
volume = "31",
pages = "265287",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz",
paper = "Gath85.ps"
}
@@ 10910,7 +10547,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Kalt87,
author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.",
 title = "Fast parallel computation of Hermite and Smith forms of polynomial matrices",
+ title = "Fast parallel computation of Hermite and Smith forms of
+ polynomial matrices",
journal = "SIAM J. Alg. Discrete Math.",
year = "1987",
volume = "8",
@@ 10941,7 +10579,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt87b,
author = "Kaltofen, E.",
 title = "Singlefactor Hensel lifting and its application to the straightline complexity of certain polynomials",
+ title = "Singlefactor Hensel lifting and its application to the
+ straightline complexity of certain polynomials",
booktitle = "Proc. 19th Annual ACM Symp. Theory Comput.",
year = "1987",
pages = "443452",
@@ 10955,7 +10594,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Kalt87c,
author = "Kaltofen, E.",
 title = "Deterministic irreducibility testing of polynomials over large finite fields",
+ title = "Deterministic irreducibility testing of polynomials over
+ large finite fields",
journal = "Journal of Symbolic Computation",
year = "1987",
volume = "4",
@@ 10969,7 +10609,9 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt88,
author = "Kaltofen, E. and Trager, B.",
 title = "Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators",
+ title = "Computing with polynomials given by black boxes for their
+ evaluations: Greatest common divisors, factorization, separation of
+ numerators and denominators",
booktitle = "Proc. 29th Annual Symp. Foundations of Comp. Sci.",
pages = "296305",
year = "1988",
@@ 10983,7 +10625,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Mill88,
author = "Miller, G.L. and Ramachandran, V. and Kaltofen, E.",
 title = "Efficient parallel evaluation of straightline code and arithmetic circuits",
+ title = "Efficient parallel evaluation of straightline code and
+ arithmetic circuits",
journal = "SIAM J. Comput.",
year = "1988",
volume = "17",
@@ 11011,7 +10654,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Kalt88b,
author = "Kaltofen, E.",
 title = "Greatest common divisors of polynomials given by straightline programs",
+ title = "Greatest common divisors of polynomials given by
+ straightline programs",
journal = "J. ACM",
year = "1988",
volume = "35",
@@ 11025,8 +10669,10 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Free88,
 author = "Freeman, T.S. and Imirzian, G. and Kaltofen, E. and Yagati, Lakshman",
 title = "DAGWOOD: A system for manipulating polynomials given by straightline programs",
+ author = "Freeman, T.S. and Imirzian, G. and Kaltofen, E. and
+ Yagati, Lakshman",
+ title = "DAGWOOD: A system for manipulating polynomials given by
+ straightline programs",
journal = "ACM Trans. Math. Software",
year = "1988",
volume = "14",
@@ 11076,7 +10722,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Kalt89a,
author = "Kaltofen, E.; Rolletschek, H.",
 title = "Computing greatest common divisors and factorizations in quadratic number fields",
+ title = "Computing greatest common divisors and factorizations in
+ quadratic number fields",
journal = "Math. Comput.",
year = "1989",
volume = "53",
@@ 11091,7 +10738,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Unpublished{Kalt89b,
author = "Kaltofen, E.",
 title = "Processor efficient parallel computation of polynomial greatest common divisors",
+ title = "Processor efficient parallel computation of polynomial greatest
+ common divisors",
year = "1989",
month = "July",
url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_gcd.ps.gz",
@@ 11108,7 +10756,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
address = "Dept. Comput. Sci., Troy, New York",
year = "1989",
month = "July",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_parallel.ps.gz",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_parallel.ps.gz",
paper = "Kalt89c.ps"
}
@@ 11172,7 +10821,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Kalt90b,
author = "Kaltofen, E.",
 title = "Computing the irreducible real factors and components of an algebraic curve",
+ title = "Computing the irreducible real factors and components of an
+ algebraic curve",
journal = "Applic. Algebra Engin. Commun. Comput.",
year = "1990",
volume = "1",
@@ 11206,7 +10856,9 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Kalt90d,
author = "Kaltofen, E.; Trager, B.",
 title = "Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators",
+ title = "Computing with polynomials given by black boxes for their
+ evaluations: Greatest common divisors, factorization, separation of
+ numerators and denominators",
journal = "J. Symbolic Comput.",
year = "1990",
volume = "9",
@@ 11240,7 +10892,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
author = "Kaltofen, E. and Singer, M.F.",
editor = "D. V. Shirkov and V. A. Rostovtsev and V. P. Gerdt",
title = "Size efficient parallel algebraic circuits for partial derivatives",
 booktitle = "IV International Conference on Computer Algebra in Physical Research",
+ booktitle =
+ "IV International Conference on Computer Algebra in Physical Research",
pages = "133145",
publisher = "World Scientific Publ. Co.",
year = "1991",
@@ 11254,8 +10907,10 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InCollection{Kalt91b,
author = "Kaltofen, E. and Yui, N.",
 editor = "D. V. Chudnovsky and G. V. Chudnovsky and H. Cohn and M. B. Nathanson",
 title = "Explicit construction of {Hilbert} class fields of imaginary quadratic fields by integer lattice reduction",
+ editor = "D. V. Chudnovsky and G. V. Chudnovsky and H. Cohn and
+ M. B. Nathanson",
+ title = "Explicit construction of {Hilbert} class fields of imaginary
+ quadratic fields by integer lattice reduction",
booktitle = "Number Theory New York Seminar 19891990",
pages = "150202",
publisher = "SpringerVerlag",
@@ 11282,7 +10937,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt91c,
author = "Kaltofen, E. and Pan, V.",
 title = "Processor efficient parallel solution of linear systems over an abstract field",
+ title = "Processor efficient parallel solution of linear systems over
+ an abstract field",
booktitle = "Proc. SPAA '91 3rd Ann. ACM Symp. Parallel Algor. Architecture",
pages = "180191",
publisher = "ACM Press",
@@ 11312,7 +10968,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt92,
author = "Kaltofen, E. and Pan, V.",
 title = "Processorefficient parallel solution of linear systems {II}: the positive characteristic and singular cases",
+ title = "Processorefficient parallel solution of linear systems {II}:
+ the positive characteristic and singular cases",
booktitle = "Proc. 33rd Annual Symp. Foundations of Comp. Sci.",
year = "1992",
pages = "714723",
@@ 11359,7 +11016,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
@InProceedings{Kalt93,
author = "Kaltofen, E.",
title = "Computational Differentiation and Algebraic Complexity Theory",
 booktitle = "Workshop Report on First Theory Institute on Computational Differentiation",
+ booktitle = "Workshop Report on First Theory Institute on Computational
+ Differentiation",
editor = "C. H. Bischof and A. Griewank and P. M. Khademi",
publisher = "Argonne National Laboratory",
address = "Argonne, Illinois",
@@ 11384,7 +11042,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
publisher = "Morgan Kaufmann Publ.",
year = "1993",
address = "San Mateo, California",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_synthesis.ps.gz",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_synthesis.ps.gz",
paper = "Kalt93a.ps"
}
@@ 11394,7 +11053,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
@InProceedings{Diaz93,
author = "Diaz, A. and Kaltofen, E. and Lobo, A. and Valente, T.",
editor = "A. Miola",
 title = "Process scheduling in {DSC} and the large sparse linear systems challenge",
+ title = "Process scheduling in {DSC} and the large sparse linear
+ systems challenge",
booktitle = "Proc. DISCO '93",
series = "Lect. Notes Comput. Sci.",
pages = "6680",
@@ 11416,7 +11076,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
volume = "27",
number = "4",
pages = "2",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_sambull.ps.gz",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_sambull.ps.gz",
paper = "Kalt93b.ps"
}
@@ 11425,7 +11086,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt94,
author = "Kaltofen, E. and Pan, V.",
 title = "Parallel solution of Toeplitz and Toeplitzlike linear systems over fields of small positive characteristic",
+ title = "Parallel solution of Toeplitz and Toeplitzlike linear
+ systems over fields of small positive characteristic",
booktitle = "Proc. First Internat. Symp. Parallel Symbolic Comput.",
crossref = "PASCO94",
pages = "225233",
@@ 11455,7 +11117,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt94a,
author = "Kaltofen, E. and Lobo, A.",
 title = "Factoring highdegree polynomials by the black box Berlekamp algorithm",
+ title = "Factoring highdegree polynomials by the black box
+ Berlekamp algorithm",
booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
crossref = "ISSAC94",
pages = "9098",
@@ 11468,7 +11131,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt94b,
author = "Kaltofen, E.",
 title = "Asymptotically fast solution of {Toeplitz}like singular linear systems",
+ title = "Asymptotically fast solution of {Toeplitz}like singular
+ linear systems",
booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
pages = "297304",
crossref = "ISSAC94",
@@ 11482,13 +11146,15 @@ relatively insensitive to the multiplicities of the squarefree factors.
@InProceedings{Sama95,
author = "Samadani, M. and Kaltofen, E.",
title = "Prediction based task scheduling in distributed computing",
 booktitle = "Languages, Compilers and RunTime Systems for Scalable Computers",
+ booktitle = "Languages, Compilers and RunTime Systems for Scalable
+ Computers",
editor = "B. K. Szymanski and B. Sinharoy",
publisher = "Kluwer Academic Publ.",
address = "Boston",
pages = "317320",
year = "1996",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/SaKa95_poster.ps.gz",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/95/SaKa95_poster.ps.gz",
paper = "Sama95.ps"
}
@@ 11497,7 +11163,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Kalt95,
author = "Kaltofen, E.",
 title = "Analysis of {Coppersmith}'s block {Wiedemann} algorithm for the parallel solution of sparse linear systems",
+ title = "Analysis of {Coppersmith}'s block {Wiedemann} algorithm for the
+ parallel solution of sparse linear systems",
journal = "Math. Comput.",
year = "1995",
volume = "64",
@@ 11512,7 +11179,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Diaz95,
author = "Diaz, A. and Kaltofen, E.",
 title = "On computing greatest common divisors with polynomials given by black boxes for their evaluation",
+ title = "On computing greatest common divisors with polynomials given by
+ black boxes for their evaluation",
booktitle = "Proc. 1995 Internat. Symp. Symbolic Algebraic Comput.",
crossref = "ISSAC95",
pages = "232239",
@@ 11554,8 +11222,10 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@Article{Diaz95a,
 author = "Diaz, A. and Hitz, M. and Kaltofen, E. and Lobo, A. and Valtente, T.",
 title = "Process scheduling in {DSC} and the large sparse linear systems challenge",
+ author = "Diaz, A. and Hitz, M. and Kaltofen, E. and Lobo, A. and
+ Valtente, T.",
+ title = "Process scheduling in {DSC} and the large sparse linear
+ systems challenge",
journal = "Journal of Symbolic Computing",
year = "1995",
volume = "19",
@@ 11611,7 +11281,8 @@ relatively insensitive to the multiplicities of the squarefree factors.
\begin{chunk}{axiom.bib}
@InProceedings{Kalt96a,
author = "Kaltofen, E. and Lobo, A.",
 title = "Distributed matrixfree solution of large sparse linear systems over finite fields",
+ title = "Distributed matrixfree solution of large sparse linear systems
+ over finite fields",
booktitle = "Proc. High Performance Computing '96",
year = "1996",
editor = "A. M. Tentner",
@@ 11629,13 +11300,15 @@ relatively insensitive to the multiplicities of the squarefree factors.
@InProceedings{Kalt96b,
author = "Kaltofen, E.",
title = "Blocked iterative sparse linear system solvers for finite fields",
 booktitle = "Proc. Symp. Parallel Comput. Solving Large Scale Irregular Applic. (Stratagem '96)",
+ booktitle = "Proc. Symp. Parallel Comput. Solving Large Scale Irregular
+ Applic. (Stratagem '96)",
editor = "C. Roucairol",
publisher = "INRIA",
address = "Sophia Antipolis, France",
pages = "9195",
year = "1996",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/Ka96_stratagem.ps.gz",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/96/Ka96_stratagem.ps.gz",
paper = "Kalt96b.ps"
}
@@ 11653,24 +11326,23 @@ relatively insensitive to the multiplicities of the squarefree factors.
note = "Special issue on education, L. Lambe, editor.",
url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/Ka97_jsc.pdf",
keywords = "axiomref,read",
 paper = "Kalt97.pdf"
+ paper = "Kalt97.pdf",
+ abstract = "
+ We report on the contents and pedagogy of a course in abstract algebra
+ that was taught with the aid of educational software developed within
+ the Mathematica system. We describe the topics covered and the
+ didactical use of the corresponding Mathematica packages, as well as
+ draw conclusions for future such courses from the students' comments
+ and our own experience."
}
\end{chunk}
\begin{adjustwidth}{2.5em}{0pt}
We report on the contents and pedagogy of a course in abstract algebra
that was taught with the aid of educational software developed within
the Mathematica system. We describe the topics covered and the
didactical use of the corresponding Mathematica packages, as well as
draw conclusions for future such courses from the students' comments
and our own experience.
\end{adjustwidth}

\begin{chunk}{axiom.bib}
@InProceedings{Kalt97a,
author = "Kaltofen, E. and Shoup, V.",
 title = "Fast polynomial factorization over high algebraic extensions of finite fields",
+ title = "Fast polynomial factorization over high algebraic extensions of
+ finite fields",
booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.",
crossref = "ISSAC97",
pages = "184188",
diff git a/changelog b/changelog
index a6c878c..14f3319 100644
 a/changelog
+++ b/changelog
@@ 1,3 +1,5 @@
+20140920 tpd src/axiomwebsite/patches.html 20140920.01.tpd.patch
+20140920 tpd books/bookvolbib add abstracts, rearrange, add new entries
20140919 tpd src/axiomwebsite/patches.html 20140919.01.tpd.patch
20140919 tpd books/axiom.bst use axiom specific bib style
20140919 tpd books/Makefileuse axiom.bst for bib style
diff git a/patch b/patch
index 22d4a58..a644f10 100644
 a/patch
+++ b/patch
@@ 1,3 +1,3 @@
books/axiom.bst use axiom specific bib style
+books/bookvolbib add abstracts, rearrange, add new entries
All of the books now use an axiomspecific bibtex format for the biblography.
+Expand and cleanup bibliography
diff git a/src/axiomwebsite/patches.html b/src/axiomwebsite/patches.html
index 7a7c5e0..5e0a8d3 100644
 a/src/axiomwebsite/patches.html
+++ b/src/axiomwebsite/patches.html
@@ 4646,6 +4646,8 @@ books/bookvol*pamphlet rebuild Axiom using bibtex
books/bookvolbib add references
20140919.01.tpd.patch
books/axiom.bst use axiom specific bib style
+20140920.01.tpd.patch
+books/bookvolbib add abstracts, rearrange, add new entries