diff git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet
index eb8d61f..600363f 100644
 a/books/bookvolbib.pamphlet
+++ b/books/bookvolbib.pamphlet
@@ 10,11351 +10,12439 @@ 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{Special Topics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Solving Systems of Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Linear Algebra} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
@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."
+@Unpublished{Kalt01,
+ author = "Kaltofen, E.",
+ title = "Algorithms for sparse and black box matrices
+ over finite fields (Invited talk)",
+ year = "2001",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/Ka01_Fq6.pdf",
+ paper = "Kalt01.pdf"
}
\end{chunk}
\subsection{Numerical Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
{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."
+\begin{chunk}{axiom.bib}
+@Article{Chen02,
+ author = "Chen, L. and Eberly, W. and Kaltofen, E.
+ and Saunders, B. D. and Turner, W. J. and Villard, G.",
+ title = "Efficient Matrix Preconditioners for Black Box Linear Algebra",
+ journal = "Linear Algebra and Applications",
+ year = "2002",
+ volume = "343344",
+ pages = "119146",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/CEKSTV02.pdf",
+ paper = "Chen02.pdf"
}
\end{chunk}
\begin{chunk}{ignore}
{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."
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt11d,
+ author = "Kaltofen, Erich and Storjohann, Arne",
+ title = "The Complexity of Computational Problems in Exact Linear Algebra",
+ booktitle = "Encyclopedia of Applied and Computational Mathematics",
+ crossref = "EACM",
+ year = "2011",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KS11.pdf",
+ paper = "Kalt11d.pdf"
}
\end{chunk}
\begin{chunk}{ignore}
{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"
+\begin{chunk}{axiom.bib}
+@Article{Come12,
+ author = "Comer, Matthew T. and Kaltofen, Erich L.",
+ title = "On the {Berlekamp}/{Massey} Algorithm and Counting Singular {Hankel}
+ Matrices over a Finite Field",
+ year = "2012",
+ month = "April",
+ journal = "Journal of Symbolic Computation",
+ volume = "47",
+ number = "4",
+ pages = "480491",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/CoKa10.pdf",
+ paper = "Come12.pdf"
}
\end{chunk}
\subsection{Special Functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
{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."
+\begin{chunk}{axiom.bib}
+@Article{Kalt13a,
+ author = "Kaltofen, Erich and Yuhasz, George",
+ title = "A Fraction Free Matrix {Berlekamp}/{Massey} Algorithm",
+ journal = "Linear Algebra and Applications",
+ year = "2013",
+ volume = "439",
+ number = "9",
+ month = "November",
+ pages = "25152526",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/08/KaYu08.pdf",
+ paper = "Kalt13a.pdf"
}
\end{chunk}
\begin{chunk}{ignore}
{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."
+\begin{chunk}{axiom.bib}
+@Article{Kalt13,
+ author = "Kaltofen, Erich and Yuhasz, George",
+ title = "On The Matrix {Berlekamp}{Massey} Algorithm",
+ year = "2013",
+ volume = "9",
+ number = "4",
+ month = "September",
+ journal = "ACM Trans. Algorithms",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KaYu06.pdf",
+ paper = "Kalt13.pdf"
}
\end{chunk}
\subsection{Exponential Integral $E_1(x)$} %%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
{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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt02a,
+ author = "Kaltofen, Erich",
+ title = "An outputsensitive variant of the baby steps/\allowbreak
+ giant steps determinant algorithm",
+ booktitle = "Proc. 2002 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC02",
+ pages = "138144",
+ year = "2002",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/Ka02.pdf",
+ paper = "Kalt02a.pdf"
}
\end{chunk}
\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",
 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."
+@InProceedings{Kalt01a,
+ author = "Kaltofen, E. and Villard, G.",
+ title = "On the complexity of computing determinants",
+ booktitle = "Proc. Fifth Asian Symposium on Computer Mathematics
+ (ASCM 2001)",
+ crossref = "ASCM01",
+ pages = "1327",
+ isbn = "981024763X",
+ year = "2001",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/KaVi01.pdf",
+ paper = "Kalt01a.pdf"
}
\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."
+@Article{Kalt04a,
+ author = "Kaltofen, Erich and Villard, Gilles",
+ title = "On the Complexity of Computing Determinants",
+ journal = "Computational Complexity",
+ volume = "13",
+ number = "34",
+ year = "2004",
+ pages = "91130",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/04/KaVi04_2697263.pdf",
+ paper = "Kalt04a.pdf"
}
\end{chunk}
\subsection{Polynomial GCD} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt97b,
+ author = "Eberly, W. and Kaltofen, E.",
+ title = "On Randomized {Lanczos} Algorithms",
+ booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.",
+ year = "1997",
+ crossref = "ISSAC97",
+ pages = "176183",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/EbKa97.pdf",
+ paper = "Kalt97b.pdf"
\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
+}
\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
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt94b,
+ author = "Kaltofen, E.",
+ title = "Asymptotically fast solution of {Toeplitz}like singular
+ linear systems",
+ booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
+ pages = "297304",
+ crossref = "ISSAC94",
+ year = "1994",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/Ka94_issac.pdf",
+ paper = "Kalt94b.pdf"
+}
\end{chunk}
\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
 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."
+\begin{chunk}{axiom.bib}
+@Article{Kalt99,
+ author = "Kaltofen, E. and Lobo, A",
+ title = "Distributed matrixfree solution of large sparse linear systems over
+ finite fields",
+ journal = "Algorithmica",
+ year = "1999",
+ pages = "331348",
+ month = "JulyAug.",
+ volume = "24",
+ number = "34",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/KaLo99.pdf",
+ paper = "Kalt99.pdf"
+}
\end{chunk}
\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
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt96a,
+ author = "Kaltofen, E. and Lobo, A.",
+ title = "Distributed matrixfree solution of large sparse linear systems
+ over finite fields",
+ booktitle = "Proc. High Performance Computing '96",
+ year = "1996",
+ editor = "A. M. Tentner",
+ pages = "244247",
+ organization = "Society for Computer Simulation",
+ publisher = "Simulation Councils, Inc.",
+ address = "San Diego, CA",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_hpc.pdf",
+ paper = "Kalt96a.pdf"
+}
\end{chunk}
\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",
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt94a,
+ author = "Kaltofen, E. and Lobo, A.",
+ title = "Factoring highdegree polynomials by the black box
+ Berlekamp algorithm",
+ booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC94",
+ pages = "9098",
+ year = "1994",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaLo94.ps.gz",
+ paper = "Kalt94a.ps"
+}
\end{chunk}
\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
 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."
+\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",
+ journal = "Math. Comput.",
+ year = "1995",
+ volume = "64",
+ number = "210",
+ pages = "777806",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_mathcomp.pdf",
+ paper = "Kalt95.pdf"
+}
\end{chunk}
\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."
+\begin{chunk}{axiom.bib}
+@Article{Kalt90a,
+ author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.",
+ title = "Parallel algorithms for matrix normal forms",
+ journal = "Linear Algebra and Applications",
+ year = "1990",
+ volume = "136",
+ pages = "189208",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KKS90.pdf",
+ paper = "Kalt90a.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wiki 4]{Wiki4}.
``Polynomial greatest common divisor''
\verben.wikipedia.org/wiki/Polynomial_greatest_common_divisor
+\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",
+ journal = "SIAM J. Alg. Discrete Math.",
+ year = "1987",
+ volume = "8",
+ pages = "683690",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/KKS87.pdf",
+ paper = "Kalt87.pdf"
+}
\end{chunk}
\subsection{Category Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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
 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."
+\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",
+ booktitle = "Proc. 33rd Annual Symp. Foundations of Comp. Sci.",
+ year = "1992",
+ pages = "714723",
+ publisher = "IEEE Computer Society Press",
+ address = "Los Alamitos, California",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/KaPa92.pdf",
+ paper = "Kalt92.pdf"
+}
\end{chunk}
\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
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt91c,
+ author = "Kaltofen, E. and Pan, V.",
+ 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",
+ year = "1991",
+ address = "New York, N.Y.",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaPa91.pdf",
+ paper = "Kalt91c.pdf"
+}
\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
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt91,
+ author = "Kaltofen, E. and Saunders, B.D.",
+ editor = "H. F. Mattson and T. Mora and T. R. N. Rao",
+ title = "On {Wiedemann's} method of solving sparse linear systems",
+ booktitle = "Proc. AAECC9",
+ series = "Lect. Notes Comput. Sci.",
+ volume = "539",
+ pages = "2938",
+ publisher = "SpringerVerlag",
+ year = "1991",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSa91.pdf",
+ paper = "Kalt91.pdf"
+}
\end{chunk}
\subsection{Proving Axiom Correct} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Algebraic Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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."
+\begin{chunk}{axiom.bib}
+@InCollection{Diaz97,
+ author = "Diaz, A. and Kaltofen, E. and Pan, V.",
+ title = "Algebraic Algorithms",
+ booktitle = "The Computer Science and Engineering Handbook",
+ publisher = "CRC Press",
+ year = "1997",
+ editor = "A. B. Tucker",
+ pages = "226248",
+ address = "Boca Raton, Florida",
+ chapter = "10",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/DKP97.ps.gz",
+ paper = "Diaz97.ps"
+}
\end{chunk}
\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
 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."
+\begin{chunk}{axiom.bib}
+@InCollection{Diaz99,
+ author = "Diaz, A. and Emiris, I. and Kaltofen, E. and Pan, V.",
+ title = "Algebraic Algorithms",
+ booktitle = "Algorithms \& Theory of Computation Handbook",
+ publisher = "CRC Press",
+ year = "1999",
+ editor = "M. J. Atallah",
+ address = "Boca Raton, Florida",
+ pages = "16.116.27",
+ isbn = "0849326494",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/DEKP99.ps.gz",
+ paper = "Diaz99.ps"
+}
\end{chunk}
\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."
+@InCollection{Kalt87a,
+ author = "Kaltofen, E.",
+ editor = "J. F. Traub",
+ title = "Computer algebra algorithms",
+ booktitle = "Annual Review in Computer Science",
+ pages = "91118",
+ publisher = "Annual Reviews Inc.",
+ year = "1987",
+ volume = "2",
+ address = "Palo Alto, California",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_annrev.pdf",
+ paper = "Kalt87a.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
 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.
+\section{Sparse Linear Systems} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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}{axiom.bib}
+@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)",
+ editor = "C. Roucairol",
+ publisher = "INRIA",
+ address = "Sophia Antipolis, France",
+ pages = "9195",
+ year = "1996",
+ keywords = "survey",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/96/Ka96_stratagem.ps.gz",
+ paper = "Kalt96b.ps"
+}
 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}
+
+\section{Matrix Determinants} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@Article{Kalt04,
+ author = "Kaltofen, E. and Villard, G.",
+ title = "Computing the sign or the value of the determinant of an integer
+ matrix, a complexity survey",
+ journal = "J. Computational Applied Math.",
+ volume = "162",
+ number = "1",
+ month = "January",
+ pages = "133146",
+ year = "2004",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/KaVi02.pdf",
+ paper = "Kalt04.pdf"
+}
\end{chunk}
\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."
+\section{Open Problems} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@Article{Kalt00,
+ author = "Kaltofen, E.",
+ title = "Challenges of Symbolic Computation My Favorite Open Problems",
+ journal = "Journal of Symbolic Computation",
+ volume = "29",
+ number = "6",
+ pages = "891919",
+ year = "2000",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/2K/Ka2K.pdf",
+ paper = "Kalt00.pdf"
+}
\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
 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."
+\section{Parallel Evaluation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt93a,
+ author = "Kaltofen, E.",
+ editor = "J. Reif",
+ title = "Dynamic parallel evaluation of computation {DAG}s",
+ booktitle = "Synthesis of Parallel Algorithms",
+ pages = "723758",
+ publisher = "Morgan Kaufmann Publ.",
+ year = "1993",
+ address = "San Mateo, California",
+ keywords = "survey",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_synthesis.ps.gz",
+ paper = "Kalt93a.ps"
+}
\end{chunk}
\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
 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.
+\section{Hybrid Symbolic/Numeric} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt06,
+ author = "Kaltofen, Erich and Zhi, Lihong",
+ title = "Hybrid SymbolicNumeric Computation",
+ year = "2006",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'06",
+ crossref = "ISSAC06",
+ pages = "7",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KaZhi06.pdf",
+ paper = "Kalt06.pdf"
+}
\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
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Hutt10,
+ author = "Hutton, Sharon E. and Kaltofen, Erich L. and Zhi, Lihong",
+ title = "Computing the radius of positive semidefiniteness of a
+ multivariate real polynomial via a dual of {Seidenberg}'s method",
+ year = "2010",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'10",
+ crossref = "ISSAC10",
+ pages = "227234",
+ month = "July",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/HKZ10.pdf",
+ paper = "Hutt10.pdf"
+}
\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
 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.
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt09,
+ author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong",
+ title = "A Proof of the {Monotone Column Permanent (MCP) Conjecture} for
+ Dimension 4 via SumsOfSquares of Rational Functions",
+ year = "2009",
+ booktitle = "Proc. 2009 Internat. Workshop on SymbolicNumeric Comput.",
+ crossref = "SNC09",
+ pages = "6569",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/09/KYZ09.pdf",
+ paper = "Kalt09.pdf"
+}
 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.
+\end{chunk}
 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."
+\begin{chunk}{axiom.bib}
+@Article{Kalt12,
+ author = "Kaltofen, Erich L. and Li, Bin and Yang, Zhengfeng and
+ Zhi, Lihong",
+ title = "Exact Certification in Global Polynomial Optimization
+ Via SumsOfSquares of Rational Functions
+ with Rational Coefficients",
+ year = "2012",
+ month = "January",
+ journal = "Journal of Symbolic Computation",
+ volume = "47",
+ number = "1",
+ pages = "115",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/09/KLYZ09.pdf",
+ paper = "Kalt12.pdf"
+}
\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
 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.
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt08b,
+ author = "Kaltofen, Erich and Li, Bin and Yang, Zhengfeng and Zhi, Lihong",
+ title = "Exact Certification of Global Optimality of Approximate
+ Factorizations Via Rationalizing SumsOfSquares
+ with Floating Point Scalars",
+ year = "2008",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'08",
+ crossref = "ISSAC08",
+ pages = "155163",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/08/KLYZ08.pdf",
+ paper = "Kalt08b.pdf"
+}
 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}{axiom.bib}
+@InProceedings{Kalt06b,
+ author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong",
+ title = "Approximate greatest common divisors of several polynomials
+ with linearly constrained coefficients and singular polynomials",
+ year = "2006",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'06",
+ crossref = "ISSAC06",
+ pages = "169176",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KYZ06.pdf",
+ paper = "Kalt06b.pdf"
+}
\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."
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt05,
+ author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong",
+ title = "Structured Low Rank Approximation of a {Sylvester} Matrix",
+ booktitle = "SymbolicNumeric Computation",
+ crossref = "SNC06",
+ pages = "6983",
+ year = "2005",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/05/KYZ05.pdf",
+ paper = "Kalt05.pdf"
+}
\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
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt03a,
+ author = "Kaltofen, Erich and May, John",
+ title = "On Approximate Irreducibility of Polynomials in Several Variables",
+ year = "2003",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'03",
+ crossref = "ISSAC03",
+ pages = "161168",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/KM03.pdf",
+ paper = "Kalt03a.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
\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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Gao04a,
+ author = "Shuhong, Gao and Kaltofen, Erich and May, John P. and
+ Yang, Zhengfeng and Zhi, Lihong",
+ title = "Approximate factorization of multivariate polynomials via
+ differential equations",
+ year = "2004",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'04",
+ crossref = "ISSAC04",
+ pages = "167174",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/04/GKMYZ04.pdf",
+ paper = "Gao04a.pdf"
+}
\end{chunk}
\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
 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."
+\begin{chunk}{axiom.bib}
+@Article{Kalt08,
+ author = "Kaltofen, Erich and May, John and Yang, Zhengfeng and Zhi, Lihong",
+ title = "Approximate Factorization of Multivariate Polynomials Using
+ Singular Value Decomposition",
+ year = "2008",
+ journal = "Journal of Symbolic Computation",
+ volume = "43",
+ number = "5",
+ pages = "359376",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KMYZ07.pdf",
+ paper = "Kalt08.pdf"
+}
\end{chunk}
\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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Hitz99,
+ author = "Hitz, M.A. and Kaltofen, E. and Lakshman, Y.N.",
+ title = "Efficient Algorithms for Computing the Nearest Polynomial
+ With A Real Root and Related Problems",
+ booktitle = "Proc. 1999 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC99",
+ pages = "205212",
+ year = "1999",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/HKL99.pdf",
+ paper = "Hitz99.pdf"
+}
\end{chunk}
\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
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Hitz98,
+ author = "Hitz, M. A. and Kaltofen, E.",
+ title = "Efficient Algorithms for Computing the Nearest Polynomial
+ with Constrained Roots",
+ booktitle = "Proc. 1998 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC98",
+ year = "1998",
+ pages = "236243",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/HiKa98.pdf",
+ paper = "Hitz98.pdf"
+}
\end{chunk}
+\section{Software Systems} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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",
 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."
+@InProceedings{Diaz91,
+ author = "Diaz, A.; Kaltofen,E.; Schmitz, K.; Valente, T.",
+ title = "DSC A System for Distributed Symbolic Computation",
+ booktitle = "Proc. 1991 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC91",
+ pages = "323332",
+ year = "1991",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/DKSV91.pdf",
+ paper = "Diaz91.pdf"
}
\end{chunk}
\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",
 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."
+@InProceedings{Chan94,
+ author = "Chan, K.C. and Diaz, A. and Kaltofen, E.",
+ editor = "R. J. Lopez",
+ title = "A distributed approach to problem solving in Maple",
+ booktitle = "Maple V: Mathematics and its Application",
+ pages = "1321",
+ publisher = {Birkh\"auser},
+ year = "1994",
+ series = "Proceedings of the Maple Summer Workshop and Symposium (MSWS'94)",
+ address = "Boston",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/CDK94.ps.gz",
+ paper = "Chan94.ps"
}
\end{chunk}
\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
+\begin{chunk}{axiom.bib}
+@InProceedings{Duma02,
+ author = "Dumas, J.G. and Gautier, T. and Giesbrecht, M. and Giorgi, P.
+ and Hovinen, B. and Kaltofen, E. and Saunders, B.D. and
+ Turner, W.J. and Villard, G.",
+ title = "{LinBox}: A Generic Library for Exact Linear Algebra",
+ booktitle = "Proc. First Internat. Congress Math. Software ICMS 2002,
+ Beijing, China",
+ crossref = "ICMS02",
+ pages = "4050",
+ year = "2002",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/Detal02.pdf",
+ paper = "Duma02.pdf"
+}
\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
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt05a,
+ author = "Kaltofen, Erich and Morozov, Dmitriy and Yuhasz, George",
+ title = "Generic Matrix Multiplication and Memory Management in {LinBox}",
+ year = "2005",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'05",
+ crossref = "ISSAC05",
+ pages = "216223",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/05/KMY05.pdf",
+ paper = "Kalt05a.pdf"
+}
\end{chunk}
\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",
 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."
+@InProceedings{Diaz98,
+ author = "Diaz, A. and Kaltofen, E.",
+ title = "{FoxBox}, a System for Manipulating Symbolic Objects in Black Box
+ Representation",
+ booktitle = "Proc. 1998 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC98",
+ year = "1998",
+ pages = "3037",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/DiKa98.pdf",
+ paper = "Diaz98.pdf"
}
\end{chunk}
\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
 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."
+\begin{chunk}{axiom.bib}
+@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",
+ booktitle = "Proc. DISCO '93",
+ series = "Lect. Notes Comput. Sci.",
+ pages = "6680",
+ year = "1993",
+ volume = "722",
+ publisher = "SpringerVerlag",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/DHKLV93.pdf",
+ paper = "Diaz93.pdf"
+}
\end{chunk}
\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."
+\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",
+ journal = "Journal of Symbolic Computing",
+ year = "1995",
+ volume = "19",
+ number = "13",
+ pages = "269282",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DHKLV95.pdf",
+ paper = "Diaz95a.pdf"
+}
\end{chunk}
\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
 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."
+\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",
+ journal = "ACM Trans. Math. Software",
+ year = "1988",
+ volume = "14",
+ number = "3",
+ pages = "218240",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/FIKY88.pdf",
+ paper = "Free88.pdf"
+}
\end{chunk}
\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
 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.

 The paper shows an interesting way to decorate Axiom with pre and
 postconditions.
+\section{The Seven Dwarfs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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}"
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt10a,
+ author = "Kaltofen, Erich L.",
+ title = "The ``{Seven} {Dwarfs}'' of Symbolic Computation",
+ booktitle = "Numeric and Symbolic Scientific Computing
+ Progress and Prospects",
+ crossref = "LaPau12",
+ pages = "95104",
+ year = "2010",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/Ka10_7dwarfs.pdf",
+ paper = "Kalt10a.pdf"
+}
\end{chunk}
\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
+\section{Solving Systems of Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@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 = "
 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."
+ 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{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

\end{chunk}
+\section{Numerical Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Briggs 04]{Bri04} Briggs, Keith
``Exact real arithmetic''
\verbkeithbriggs.info/documents/xrkenttalkpp.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bri04.pdf
+{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[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
+{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 = "
 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."
+ 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}{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"
+\begin{chunk}{ignore}
+{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{Numerics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Special Functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
+{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 = "
 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[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
+ 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[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
+{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 = "
 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}
@book{Hamm62,
 author = "Hamming R W.",
 title = "Numerical Methods for Scientists and Engineers",
 publisher = "Dover",
 year = "1973",
 isbn = "0486652416"
+ This is a compendium of indefinite and definite integrals of products
+ of the Error functions with elementary and transcendental functions."
}
\end{chunk}
\subsection{Advanced Documentation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Exponential Integral $E_1(x)$} %%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem [Bostock 14]{Bos14} Bostock, Mike
``Visualizing Algorithms''
\verbbost.ocks.org/mike/algorithms
+{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 website hosts various ways of visualizing algorithms. The hope is
 that these kind of techniques can be applied to Axiom."
+ This is a compendium of indefinite and definite integrals of products
+ of the Exponential Integral with elementary or transcendental functions."
+}
\end{chunk}
\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
 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[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
+\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",
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."
+ 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}{ignore}
\bibitem[Victor 11]{Vict11} Victor, Bret
``Up and Down the Ladder of Abstraction''
\verbworrydream.com/LadderOfAbstraction
+\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 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. "
+ 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{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."
+\section{Polynomial GCD} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{chunk}
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt99a,
+ author = "Kaltofen, E. and Monagan, M.",
+ title = "On the Genericity of the Modular Polynomial {GCD} Algorithm",
+ booktitle = "Proc. 1999 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC99",
+ year = "1999",
+ pages = "5966",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/KaMo99.pdf",
+ paper = "Kalt99a.pdf"
+}
\subsection{Differential Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\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[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[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
+\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 = "
 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."
+ 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[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
+\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 = "
 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."
+ 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{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
+\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 = "
 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."
+ 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{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[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 = "
 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."
+ 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{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
+\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 = "
 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."
+ 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}{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
+\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 = "
 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."
+ 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}{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
 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."
+\bibitem[Wiki 4]{Wiki4}.
+``Polynomial greatest common divisor''
+\verben.wikipedia.org/wiki/Polynomial_greatest_common_divisor
\end{chunk}
+\section{Category Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[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 = "
 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."
+ 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{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[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 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."
+ 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 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[Youssef 04]{You04} Youssef, Saul
+``Prospects for Category Theory in Aldor''
+October 2004
+%\verbaxiomdeveloper.org/axiomwebsite/papers/You04.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."
+ 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}
+\section{Proving Axiom Correct} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[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 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."
+ 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{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
+\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 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."
+ 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[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
+\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 = "
 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."
+ 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[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[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 = "
 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.
+ 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.
 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."
+ 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{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[Bertot 04]{Bert04} Bertot, Yves; Cast\'eran, Pierre
+``Interactive Theorem Proving and Program Development''
+Springer ISBN 3540208542
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}
+ 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[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
+ 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[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[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 = "
 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."
+ 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[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[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 = "
 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."
+ 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{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[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 = "
 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."
+ 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}
\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[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 = "
 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."
+ 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.
\end{chunk}
+ 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."
\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[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[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[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 = "
 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."
+ 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{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[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 = "
 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."
+ 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[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[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[Dunstan 00a]{Dun00a} Dunstan, Martin N.
+``Adding Larch/Aldor Specifications to Aldor''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dunxx.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."
+ 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[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[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 = "
 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."
+ 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{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[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 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."
+ 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[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[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 = "
 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."
+ 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}{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
+\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",
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."
+ 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}
@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{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{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[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[Bronstein 90]{Bro90b} Bronstein, Manuel
``A Unification of Liouvillian Extensions''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro90b.pdf
+\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 = "
 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."
+ 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}
@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"
+@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[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

\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",
+\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 = "
 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."
}
+ 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{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[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 = "
 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."
}
+ This is a slide deck from a talk on the correspondence between
+ Axiom/Aldor types and Logic."
\end{chunk}
\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[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 = "
 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."
+ 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[Charlwood 08]{Charl08} Charlwood, Kevin
``Symbolic Integration Problems''
\verbwww.apmaths.uwo.ca/~arich/IndependentTestResults/CharlwoodIntegrationProblems.pdf
%\verbaxiomdeveloper.org/axiomwebsite/papers/Charl08.pdf
+\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 = "
 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."
+ 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{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[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 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)."
+ 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}
+\section{Interval Arithmetic} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[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}{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
 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."
+\bibitem[Briggs 04]{Bri04} Briggs, Keith
+``Exact real arithmetic''
+\verbkeithbriggs.info/documents/xrkenttalkpp.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bri04.pdf
\end{chunk}
\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[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 = "
 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}"
+ 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}{ignore}
\bibitem[Davenport 79b]{Dav79b} Davenport, James Harold
``On the Integration of Algebraic Functions''
SpringerVerlag Lecture Notes in Computer Science 102
ISBN 0387102906
+\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"
+}
\end{chunk}
+\section{Numerics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[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 = "
 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."
+ 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[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."
+\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[Davenport 82]{Dav82} Davenport, J.H.
``On the Parallel Risch Algorithm (III): Use of Tangents''
SIGSAM V16 no. 3 pp36 August 1982
+\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}{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
 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."
+\begin{chunk}{axiom.bib}
+@book{Hamm62,
+ author = "Hamming R W.",
+ title = "Numerical Methods for Scientists and Engineers",
+ publisher = "Dover",
+ year = "1973",
+ isbn = "0486652416"
+}
\end{chunk}
+\section{Advanced Documentation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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 [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}{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",
+\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
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."
}
+ 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[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[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{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[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{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[Victor 12]{Vict12} Victor, Bret
+``Inventing on Principle''
+\verbwww.youtube.com/watch?v=PUv66718DII
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."
+ 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}
+\section{Differential Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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"
+@InProceedings{Kalt84,
+ author = "Kaltofen, E.",
+ title = "A Note on the {Risch} Differential Equation",
+ booktitle = "Proc. EUROSAM '84",
+ pages = "359366",
+ crossref = "EUROSAM84",
+ year = "1984",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_risch.ps.gz",
+ paper = "Kalt84.ps"
}
\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
 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 $O(n^4(ln nf)^2)$."
+\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[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[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 = "
 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."
+ 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}
\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[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 = "
 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."
+ 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[Kiymaz 04]{Kiym04} Kiymaz, Onur; Mirasyedioglu, Seref
``A new symbolic computation for formal integration with exact power series''
%\verbaxiomdeveloper.org/axiomwebsite/Kiym04.pdf
+\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 = "
 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."
+ 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[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[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[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[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}{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",
+\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
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."
}
+ 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[Lang 93]{Lang93} Lang, S.
``Algebra''
AddisonWesly, New York, 3rd edition 1993
+\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[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[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 = "
 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."
+ 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[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[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[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[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 = "
 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."
+ 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}{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"
}
+\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
+ 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}{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[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[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[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[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[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[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[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}{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[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{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[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 = "
 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 $Q$."
+ 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[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[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 = "
 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."
+ 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[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[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 = "
 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."
+ 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}
+\section{Expression Simplification} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[Carette 04]{Car04} Carette, Jacques
+``Understanding Expression Simplification''
+\verbwww.cas.mcmaster.ca/~carette/publications/simplification.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Car04.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."
+ 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}
\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
+\section{Integration} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@TechReport{Kalt84b,
+ author = "Kaltofen, E.",
+ title = "The Algebraic Theory of Integration",
+ institution = "RPI",
+ address = "Dept. Comput. Sci., Troy, New York",
+ year = "1984",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_integration.pdf",
+ paper = "Kalt84b.pdf"
+}
\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
 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.
+\bibitem[Adamchik xx]{Adamxx} Adamchik, Victor
+``Definite Integration''
+\verbwww.cs.cmu.edu/~adamchik/articles/integr/mj.pdf
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Adamxx.pdf
 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.
+\end{chunk}
 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."
+\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
+ 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{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[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 = "
 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."
+ 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[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[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 = "
 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."
+ 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[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[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 = "
 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."
+ 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}{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[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}
@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[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}{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[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}{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[Bronstein 90a]{Bro90a} Bronstein, Manuel
+``Integration of Elementary Functions''
+J. Symbolic Computation (1990) 9, pp117173 September 1988
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro90a.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."
}
+ 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{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"
+@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[Ritt 48]{Ritt48} Ritt, J.F.
``Integration in Finite Terms''
Columbia University Press, New York 1948
% REF:00046
+\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[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[Bronstein 90]{Bro90b} Bronstein, Manuel
+``A Unification of Liouvillian Extensions''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Bro90b.pdf
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."
+ 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{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"
+@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{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
 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.

 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{chunk}

\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
 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."
+\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
\end{chunk}
\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
+\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 = "
 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."
+ 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{chunk}
\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"
+@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[Seidenberg 58]{Sei58} Seidenberg, Abraham
``Abstract differential algebra and the analytic case''
Proc. Amer. Math. Soc. Vol 9 pp159164 (1958)
+\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{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[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[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[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 = "
 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."
+ 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{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[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[Terelius 09]{Tere09} Terelius, Bjorn
``Symbolic Integration''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Tere09.pdf
+\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 = "
 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."
+ 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{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[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 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."
}
+ 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}
\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
 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."
+\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{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",
+\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
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."
}
+ 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[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[Davenport 82a]{Dav82a} Davenport, J.H.
+``The Parallel Risch Algorithm (I)
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav82a.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."
+ 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}
\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[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{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[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[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[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{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}
+@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."
+}
\end{chunk}
\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[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}
\subsection{Ore Rings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This is used as a reference for the LeftOreRing category, in particular,
the least left common multiple (lcmCoef) function.
\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
 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."
+\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[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[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 = "
 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."
+ 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[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."
+\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}
\subsection{Number Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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[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}
\subsection{Polynomial Factorization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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
+ 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."
\subsection{Branch Cuts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\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[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}
+
+\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
+ 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[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{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}
+@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[Lang 93]{Lang93} Lang, S.
+``Algebra''
+AddisonWesly, New York, 3rd edition 1993
+
+\end{chunk}
+
+\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
+ 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[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[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}{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}{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[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{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
+
+\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)
+
+\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
+
+\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
+ 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[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{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
+ 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{chunk}{ignore}
+\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[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[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[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."
+
+\end{chunk}
+
+\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
+ 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[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}{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}{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}{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}{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{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"
+}
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\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[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}{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[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.
+
+ 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{chunk}
+
+\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
+ 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[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}{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[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[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[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[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[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}{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[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}{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[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}
+
+\section{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
+
+\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
+
+\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
+
+\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
+
+\end{chunk}
+
+\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
+
+\end{chunk}
+\section{Ore Rings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+This is used as a reference for the LeftOreRing category, in particular,
+the least left common multiple (lcmCoef) function.
+
+\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
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.
+ 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."
 In [14] an algorithm was presented for zerotesting functions defined
 by ordinary differential equations, in terms of their power series.
+\end{chunk}
 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]."
+\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
+ 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[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}
+
+\section{Number Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt89d,
+ author = "Kaltofen, E. and Valente, T. and Yui, N.",
+ title = "An improved {Las Vegas} primality test",
+ booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC89",
+ pages = "2633",
+ year = "1989",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KVY89.pdf",
+ paper = "Kalt89d.pdf"
}
\end{chunk}
\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",
+@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",
+ booktitle = "Number Theory New York Seminar 19891990",
+ pages = "150202",
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."
+ year = "1991",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaYui91.pdf",
+ paper = "Kalt91b.pdf"
+}
+
+\end{chunk}
+
+\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",
+ booktitle = "Proc. EUROSAM '84",
+ pages = "310320",
+ crossref = "EUROSAM84",
+ year = "1984",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/84/KaYui84_eurosam.ps.gz",
+ paper = "Kalt84a.ps"
+}
+
+\end{chunk}
+
+\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
+
+\end{chunk}
+
+\section{Sparse Polynomial Interpolation} %%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt07a,
+ author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong",
+ title = "On probabilistic analysis of randomization in hybrid
+ symbolicnumeric algorithms",
+ year = "2007",
+ booktitle = "Proc. 2007 Internat. Workshop on SymbolicNumeric Comput.",
+ crossref = "SNC07",
+ pages = "1117",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KYZ07.pdf",
+ paper = "Kalt07a.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt07b,
+ author = "Kaltofen, Erich and Yang, Zhengfeng",
+ title = "On Exact and Approximate Interpolation of Sparse
+ Rational Functions",
+ year = "2007",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'07",
+ crossref = "ISSAC07",
+ pages = "203210",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KaYa07.pdf",
+ paper = "Kalt07b.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@Article{Gies03,
+ author = "Giesbrecht, Mark and Kaltofen, Erich and Lee, Wenshin",
+ title = "Algorithms for Computing Sparsest Shifts of Polynomials in
+ Power, {Chebychev}, and {Pochhammer} Bases",
+ year = "2003",
+ journal = "Journal of Symbolic Computation",
+ volume = "36",
+ number = "34",
+ pages = "401424",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/GKL03.pdf",
+ paper = "Gies03.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Gies02,
+ author = "Giesbrecht, Mark and Kaltofen, Erich and Lee, Wenshin",
+ title = "Algorithms for Computing the Sparsest Shifts for Polynomials via the
+ {Berlekamp}/{Massey} Algorithm",
+ booktitle = "Proc. 2002 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC02",
+ pages = "101108",
+ year = "2002",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/GKL02.pdf",
+ paper = "Gies02.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@Article{Kalt03b,
+ author = "Kaltofen, Erich and Lee, Wenshin",
+ title = "Early Termination in Sparse Interpolation Algorithms",
+ year = "2003",
+ journal = "Journal of Symbolic Computation",
+ volume = "36",
+ number = "34",
+ pages = "365400",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/KL03.pdf",
+ paper = "Kalt03b.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt00a,
+ author = "Kaltofen, E. and Lee, W.s. and Lobo, A.A.",
+ title = "Early termination in {BenOr/Tiwari} sparse interpolation
+ and a hybrid of {Zippel}'s algorithm",
+ booktitle = "Proc. 2000 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC2K",
+ pages = "192201",
+ year = "2000",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/2K/KLL2K.pdf",
+ paper = "Kalt00a.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt10b,
+ author = "Kaltofen, Erich L.",
+ title = "Fifteen years after {DSC} and {WLSS2} {What} parallel
+ computations {I} do today [{Invited} Lecture at {PASCO} 2010]",
+ year = "2010",
+ booktitle = "Proc. 2010 Internat. Workshop on Parallel Symbolic Comput.",
+ crossref = "PASCO10",
+ pages = "1017",
+ month = "July",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/Ka10_pasco.pdf",
+ paper = "Kalt10b.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt90,
+ author = "Kaltofen, E. and Lakshman, Y.N. and Wiley, J.M.",
+ editor = "S. Watanabe and M. Nagata",
+ title = "Modular rational sparse multivariate polynomial interpolation",
+ booktitle = "Proc. 1990 Internat. Symp. Symbolic Algebraic Comput.",
+ pages = "135139",
+ publisher = "ACM Press",
+ year = "1990",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KLW90.pdf",
+ paper = "Kalt90.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt88a,
+ author = "Kaltofen, E. and Yagati, Lakshman",
+ title = "Improved sparse multivariate polynomial interpolation algorithms",
+ booktitle = "Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc.",
+ crossref = "ISSAC88",
+ pages = "467474",
+ year = "1988",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/KaLa88.pdf",
+ paper = "Kalt88a.pdf"
}

+
\end{chunk}
+\section{Divisions and Algebraic Complexity} %%%%%%%%%%%%%%%%%%%%%%%
+
\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."
+@InCollection{Gren11,
+ author = "Grenet, Bruno and Kaltofen, Erich L. and Koiran, Pascal
+ and Portier, Natacha",
+ title = "Symmetric Determinantal Representation of Formulas and Weakly
+ Skew Circuits",
+ booktitle = "Randomization, Relaxation, and Complexity in Polynomial
+ Equation Solving",
+ year = "2011",
+ editor = "Leonid Gurvits and Philippe P\'{e}bay and J. Maurice Rojas
+ and David Thompson",
+ pages = "6196",
+ publisher = "American Mathematical Society",
+ address = "Providence, Rhode Island, USA",
+ isbn = "9780821852286",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/GKKP10.pdf",
+ paper = "Gren11.pdf"
}
\end{chunk}
\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."
+@InProceedings{Kalt08a,
+ author = "Kaltofen, Erich and Koiran, Pascal",
+ title = "Expressing a Fraction of Two Determinants as a Determinant",
+ year = "2008",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'08",
+ crossref = "ISSAC08",
+ pages = "141146",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/08/KaKoi08.pdf",
+ paper = "Kalt08a.pdf"
}
\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",
 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."
+@Article{Hitz95,
+ author = "Kitz, M.A. and Kaltofen, E.",
+ title = "Integer division in residue number systems",
+ journal = "IEEE Trans. Computers",
+ year = "1995",
+ volume = "44",
+ number = "8",
+ pages = "983989",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/HiKa95.pdf",
+ paper = "Hitz95.pdf"
}

+
\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",
 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.
+@InProceedings{Kalt92a,
+ author = "Kaltofen, E.",
+ title = "On computing determinants of matrices without divisions",
+ booktitle = "Proc. 1992 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC92",
+ pages = "342349",
+ year = "1992",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_issac.pdf",
+ paper = "Kalt92a.pdf"
+}
 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}{axiom.bib}
+@Article{Cant91,
+ author = "Cantor, D.G. and Kaltofen, E.",
+ title = "On fast multiplication of polynomials over arbitrary algebras",
+ journal = "Acta Inform.",
+ year = "1991",
+ volume = "28",
+ number = "7",
+ pages = "693701",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/CaKa91.pdf",
+ paper = "Cant91.pdf"
}
\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",
 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."
+@Article{Kalt88b,
+ author = "Kaltofen, E.",
+ title = "Greatest common divisors of polynomials given by
+ straightline programs",
+ journal = "J. ACM",
+ year = "1988",
+ volume = "35",
+ number = "1",
+ pages = "231264",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/Ka88_jacm.pdf",
+ paper = "Kalt88b.pdf"
}
\end{chunk}
+\section{Polynomial Factorization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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"
+@PhdThesis{Kalt82,
+ author = "Kaltofen, E.",
+ title = "On the complexity of factoring polynomials with integer
+ coefficients",
+ school = "RPI",
+ address = "Troy, N. Y.",
+ year = "1982",
+ month = "December",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_thesis.pdf",
+ paper = "Kalt82.pdf"
}
\end{chunk}
+\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",
 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."
+@Article{Gath85,
+ author = "Gathen, Joachim von zur; Kaltofen, E.",
+ title = "Factoring sparse multivariate polynomials",
+ journal = "J. Comput. System Sci.",
+ year = "1985",
+ volume = "31",
+ pages = "265287",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz",
+ paper = "Gath85.ps"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt11c,
+ author = "Kaltofen, Erich and Lecerf, Gr{\'e}goire",
+ title = "Section 11.5. {Factorization} of multivariate polynomials",
+ booktitle = "Handbook of Finite Fields",
+ crossref = "HFF11",
+ pages = "382392",
+ year = "2011",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KL11.pdf",
+ paper = "Kalt11c.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt05b,
+ author = "Kaltofen, Erich and Koiran, Pascal",
+ title = "On the complexity of factoring bivariate supersparse
+ (lacunary) polynomials",
+ year = "2005",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'05",
+ crossref = "ISSAC05",
+ pages = "208215",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/05/KaKoi05.pdf",
+ paper = "Kalt05b.pdf"
}
\end{chunk}
\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."
+@InProceedings{Kalt06a,
+ author = "Kaltofen, Erich and Koiran, Pascal",
+ title = "Finding Small Degree Factors of Multivariate Supersparse
+ (Lacunary) Polynomials Over Algebraic Number Fields",
+ year = "2006",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'06",
+ crossref = "ISSAC06",
+ pages = "162168",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KaKoi06.pdf",
+ paper = "Kalt06a.pdf"
}
\end{chunk}
\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."
+@InProceedings{Kalt97a,
+ author = "Kaltofen, E. and Shoup, V.",
+ title = "Fast polynomial factorization over high algebraic extensions of
+ finite fields",
+ booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC97",
+ year = "1997",
+ pages = "184188",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/KaSh97.pdf",
+ paper = "Kalt97a.pdf"
}
\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",
 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."
+@Article{Kalt98,
+ author = "Kaltofen, E. and Shoup, V.",
+ title = "Subquadratictime factoring of polynomials over finite fields",
+ journal = "Math. Comput.",
+ month = "July",
+ year = "1998",
+ volume = "67",
+ number = "223",
+ pages = "11791197",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/KaSh98.pdf",
+ paper = "Kalt98.pdf"
}
\end{chunk}
\subsection{Squarefree Decomposition } %%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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.

 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
+@InProceedings{Kalt95a,
+ author = "Kaltofen, E. and Shoup, V.",
+ title = "Subquadratictime factoring of polynomials over finite fields",
+ booktitle = "Proc. 27th Annual ACM Symp. Theory Comput.",
+ year = "1995",
+ publisher = "ACM Press",
+ address = "New York, N.Y.",
+ pages = "398406",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/KaSh95.ps.gz",
+ paper = "Kalt95a.ps"
+}
 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.
+\end{chunk}
 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."
+\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",
+ booktitle = "Proc. 1995 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC95",
+ pages = "232239",
+ year = "1995",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DiKa95.ps.gz",
+ paper = "Diaz95.ps"
}
\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",
 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."
+@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",
+ booktitle = "Proc. 29th Annual Symp. Foundations of Comp. Sci.",
+ pages = "296305",
+ year = "1988",
+ organization = "IEEE",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/focs88.ps.gz",
+ paper = "Kalt88.ps"
}
\end{chunk}
\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."
+@InProceedings{Kalt85b,
+ author = "Kaltofen, E.",
+ title = "Computing with polynomials given by straightline programs {II};
+ sparse factorization",
+ booktitle = "Proc. 26th Annual Symp. Foundations of Comp. Sci.",
+ year = "1985",
+ pages = "451458",
+ organization = "IEEE",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_focs.ps.gz",
+ paper = "Kalt85b.ps"
}
\end{chunk}
\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."
+@InProceedings{Kalt86,
+ author = "Kaltofen, E.",
+ title = "Uniform closure properties of pcomputable functions",
+ booktitle = "Proc. 18th Annual ACM Symp. Theory Comput.",
+ year = "1986",
+ pages = "330337",
+ organization = "ACM",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/86/Ka86_stoc.pdf",
+ paper = "Kalt86.pdf"
}
\end{chunk}
\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."
+@InProceedings{Kalt87b,
+ author = "Kaltofen, E.",
+ 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",
+ organization = "ACM",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_stoc.pdf",
+ paper = "Kalt87b.pdf"
}
\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"
+@InCollection{Kalt89,
+ author = "Kaltofen, E.",
+ editor = "S. Micali",
+ title = "Factorization of polynomials given by straightline programs",
+ booktitle = "Randomness and Computation",
+ pages = "375412",
+ publisher = "JAI Press Inc.",
+ year = "1989",
+ volume = "5",
+ series = "Advances in Computing Research",
+ address = "Greenwhich, Connecticut",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_slpfac.pdf",
+ paper = "Kalt89.pdf"
}
\end{chunk}
\section{Axiom Citations in the Literature}

\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{axiom.bib}
+@Article{Gao04,
+ author = "Gao, Shuhong and Kaltofen, E. and Lauder, A.",
+ title = "Deterministic distinct degree factorization for polynomials
+ over finite fields",
+ year = "2004",
+ journal = "Journal of Symbolic Computation",
+ volume = "38",
+ number = "6",
+ pages = "14611470",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/GKL01.pdf",
+ paper = "Gao04.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@Article{Kalt87c,
+ author = "Kaltofen, E.",
+ title = "Deterministic irreducibility testing of polynomials over
+ large finite fields",
+ journal = "Journal of Symbolic Computation",
+ year = "1987",
+ volume = "4",
+ pages = "7782",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_jsc.ps.gz",
+ paper = "Kalt87c.ps"
+}
\end{chunk}
\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"
+@Article{Kalt95b,
+ author = "Kaltofen, E.",
+ title = "Effective {Noether} irreducibility forms and applications",
+ journal = "J. Comput. System Sci.",
+ year = "1995",
+ volume = "50",
+ number = "2",
+ pages = "274295",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_jcss.pdf",
+ paper = "Kalt95b.pdf"
}
\end{chunk}
\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",
+\begin{chunk}{axiom.bib}
+@Article{Kalt85a,
+ author = "Kaltofen, E.",
+ title = "Fast parallel absolute irreducibility testing",
+ journal = "Journal of Symbolic Computation",
+ year = "1985",
+ volume = "1",
+ number = "1",
+ pages = "5767",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_jsc.pdf",
+ paper = "Kalt85a.pdf"
+}
\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]

\end{chunk}
+\begin{chunk}{axiom.bib}
+@Article{Gath85a,
+ author = "{von zur Gathen}, Joachim and Kaltofen, E.",
+ title = "Factoring multivariate polynomials over finite fields",
+ journal = "Math. Comput.",
+ year = "1985",
+ volume = "45",
+ pages = "251261",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz",
+ paper = "Gath85a.ps"
+}
\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
+\begin{chunk}{axiom.bib}
+@Article{Kalt85e,
+ author = "Kaltofen, E.",
+ title = "Polynomialtime reductions from multivariate to bi and univariate
+ integral polynomial factorization",
+ journal = "{SIAM} J. Comput.",
+ year = "1985",
+ volume = "14",
+ number = "2",
+ pages = "469489",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_sicomp.pdf",
+ paper = "Kalt85e.pdf"
+}
\end{chunk}
\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.
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt82a,
+ author = "Kaltofen, E.",
+ title = "A polynomialtime reduction from bivariate to univariate
+ integral polynomial factorization",
+ booktitle = "Proc. 23rd Annual Symp. Foundations of Comp. Sci.",
+ year = "1982",
+ pages = "5764",
+ organization = "IEEE",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_focs.pdf",
+ paper = "Kalt82a.pdf"
+}
\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

\end{chunk}
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt03,
+ author = "Kaltofen, Erich",
+ title = "Polynomial Factorization: a Success Story",
+ year = "2003",
+ booktitle = "Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc.",
+ crossref = "ISSAC03",
+ pages = "34",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/Ka03.pdf",
+ keywords = "survey",
+ paper = "Kalt03.pdf"
+}
\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
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt92b,
+ author = "Kaltofen, E.",
+ title = "Polynomial factorization 19871991",
+ booktitle = "Proc. LATIN '92",
+ editor = "I. Simon",
+ series = "Lect. Notes Comput. Sci.",
+ volume = "583",
+ pages = "294313",
+ publisher = "SpringerVerlag",
+ year = "1992",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_latin.pdf",
+ keywords = "survey",
+ paper = "Kalt92b.pdf"
+}
\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",
 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}"
+@InCollection{Kalt90c,
+ author = "Kaltofen, E.",
+ editor = "D. V. Chudnovsky and R. D. Jenks",
+ title = "Polynomial Factorization 19821986",
+ booktitle = "Computers in Mathematics",
+ pages = "285309",
+ publisher = "Marcel Dekker, Inc.",
+ year = "1990",
+ volume = "125",
+ series = "Lecture Notes in Pure and Applied Mathematics",
+ address = "New York, N. Y.",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_survey.ps.gz",
+ keywords = "survey",
+ paper = "Kalt90c.ps"
}
\end{chunk}
\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,
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt82b,
+ author = "Kaltofen, E.",
+ title = "Polynomial factorization",
+ editor = "B. Buchberger and G. Collins and R. Loos",
+ booktitle = "Computer Algebra",
+ edition = "2",
+ pages = "95113",
+ publisher = "SpringerVerlag",
+ year = "1982",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_survey.ps.gz",
+ keywords = "survey",
+ paper = "Kalt82b.ps"
+}
\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

\end{chunk}
+\section{Branch Cuts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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",
+@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 = "
 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.
+ 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.
 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."
+ 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{chunk}
\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",
+@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 = "
 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."
+ 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}{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",
+@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 = "
 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."
+ 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[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",
+\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}{axiom.bib}
@misc{Bou93a,
 author = "Boulanger, JeanLouis",
 title = "Axiom, language fonctionnel \`a d\'evelopement objet",
 year = "1993",
 month = "October",
 paper = "Bou93a.pdf",
 keywords = "axiomref"
+@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}
\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",
+@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 = "
 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."
+ 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.
+
+ 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[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",
+\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 = "
 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."
+ 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[Bronstein 87]{Bro87}
 author = "Bronstein, Manuel",
 title = "Integration of Algebraic and Mixed Functions",
 year = "1987",
in [Wit87], p18
 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[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",
+\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 = "
 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."
+ 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}{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",
+@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 = "
 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."
+ 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[Bronstein 91c]{Bro91c}
 author = "Bronstein, Manuel",
 title = "Computer Algebra and Indefinite Integrals",
+\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",
 paper = "Bro91c.pdf",
in Computer Aided Proofs in Analysis, K.R. Meyers et al. (eds)
SpringerVerlag, NY (1991)
 keywords = "axiomref",
+ 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 = "
 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."
+ 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[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[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",
+\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}
\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",
+\section{Squarefree Decomposition } %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\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 = "
 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."
+ 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[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",
+ 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}{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",
+@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 = "
 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."
+ 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[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{chunk}{ignore}
\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",
+\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 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."
+ 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{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"
+\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."
+}
\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"
+@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{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",
 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."
+@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}
\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",
+\section{To Be Classified} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt83,
+ author = "Kaltofen, E.",
+ title = "On the complexity of finding short vectors in integer lattices",
+ booktitle = "Proc. EUROCAL '83",
+ series = "Lect. Notes Comput. Sci.",
+ year = "1983",
+ volume = "162",
+ pages = "236244",
+ publisher = "SpringerVerlag",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/83/Ka83_eurocal.pdf",
+ paper = "Kalt83.pdf"
+}
\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"
+@InProceedings{Kalt85,
+ author = "Kaltofen, E.",
+ title = "Effective {Hilbert} Irreducibility",
+ booktitle = "Proc. EUROSAM '84",
+ pages = "275284",
+ crossref = "EUROSAM84",
+ year = "1985",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_infcontr.ps.gz",
+ paper = "Kalt85.ps"
}

\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",
 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}{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"
+@TechReport{Kalt85c,
+ author = "Kaltofen, E.",
+ title = "Sparse Hensel lifting",
+ institution = "RPI",
+ address = "Dept. Comput. Sci., Troy, N. Y.",
+ year = "1985",
+ number = "8512",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_techrep.pdf",
+ paper = "Kalt85c.pdf"
}
\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

\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
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt85d,
+ author = "Kaltofen, E.",
+ title = "Sparse Hensel lifting",
+ booktitle = "EUROCAL 85 European Conf. Comput. Algebra Proc. Vol. 2",
+ crossref = "EUROCAL85",
+ pages = "417",
+ year = "1985",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_eurocal.pdf",
+ paper = "Kalt85d.pdf"
+}
\end{chunk}
\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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",
+\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",
+ journal = "SIAM J. Comput.",
+ year = "1988",
+ volume = "17",
+ number = "4",
+ pages = "687695",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/MRK88.pdf",
+ paper = "Mill88.pdf"
+}
\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
+\begin{chunk}{axiom.bib}
+@Article{Greg88,
+ author = "Gregory, B.; Kaltofen, E.",
+ title = "Analysis of the binary complexity of asymptotically fast
+ algorithms for linear system solving",
+ journal = "SIGSAM Bulletin",
+ year = "1988",
+ month = "April",
+ volume = "22",
+ number = "2",
+ pages = "4149",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/GrKa88.pdf",
+ paper = "Grey88.pdf"
+}
\end{chunk}
\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",
+\begin{chunk}{axiom.bib}
+@Article{Kalt89a,
+ author = "Kaltofen, E.; Rolletschek, H.",
+ title = "Computing greatest common divisors and factorizations in
+ quadratic number fields",
+ journal = "Math. Comput.",
+ year = "1989",
+ volume = "53",
+ number = "188",
+ pages = "697720",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KaRo89.pdf",
+ paper = "Kalt89a.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"
+@Unpublished{Kalt89b,
+ author = "Kaltofen, E.",
+ 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",
+ paper = "Kalt89b.ps"
}
\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"
+@TechReport{Kalt89c,
+ author = "Kaltofen, E.",
+ title = "Parallel Algebraic Algorithm Design",
+ institution = "RPI",
+ address = "Dept. Comput. Sci., Troy, New York",
+ year = "1989",
+ month = "July",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_parallel.ps.gz",
+ paper = "Kalt89c.ps"
}
\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"
+@InProceedings{Cann89,
+ author = "Canny, J. and Kaltofen, E. and Yagati, Lakshman",
+ title = "Solving systems of nonlinear polynomial equations faster",
+ booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC89",
+ pages = "121128",
+ year = "1989",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/CKL89.pdf",
+ paper = "Cann89.pdf"
}
\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"
+@Article{Kalt90b,
+ author = "Kaltofen, E.",
+ title = "Computing the irreducible real factors and components of an
+ algebraic curve",
+ journal = "Applic. Algebra Engin. Commun. Comput.",
+ year = "1990",
+ volume = "1",
+ number = "2",
+ pages = "135148",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_aaecc.pdf",
+ paper = "Kalt90b.pdf"
}
\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]
+\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",
+ journal = "J. Symbolic Comput.",
+ year = "1990",
+ volume = "9",
+ number = "3",
+ pages = "301320",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KaTr90.pdf",
+ paper = "Kalt90d.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
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt91a,
+ 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",
+ pages = "133145",
+ publisher = "World Scientific Publ. Co.",
+ year = "1991",
+ address = "Singapore",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSi91.pdf",
+ paper = "Kalt91a.pdf"
+}
\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"
+@InProceedings{Kalt93,
+ author = "Kaltofen, E.",
+ title = "Computational Differentiation and Algebraic Complexity Theory",
+ 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",
+ series = "Tech. Rep.",
+ volume = "ANL/MCSTM183",
+ month = "December",
+ year = "1993",
+ pages = "2830",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_diff.pdf",
+ paper = "Kalt93.pdf"
}
\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"
+@Article{Kalt93b,
+ author = "Kaltofen, E.",
+ title = "Direct proof of a theorem by Kalkbrener, Sweedler, and Taylor",
+ journal = "SIGSAM Bulletin",
+ year = "1993",
+ volume = "27",
+ number = "4",
+ pages = "2",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_sambull.ps.gz",
+ paper = "Kalt93b.ps"
}
\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",
+\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",
+ booktitle = "Proc. First Internat. Symp. Parallel Symbolic Comput.",
+ crossref = "PASCO94",
+ pages = "225233",
+ year = "1994",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaPa94.pdf",
+ paper = "Kalt94.pdf"
+}
\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",
 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."
+\begin{chunk}{axiom.bib}
+@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",
+ 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",
+ paper = "Sama95.ps"
+}
\end{chunk}
\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."
+@InProceedings{Erli96,
+ author = "Erlingsson, U. and Kaltofen, E. and Musser, D.",
+ title = "Generic {Gram}{Schmidt} Orthogonalization by Exact Division",
+ booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC96",
+ year = "1996",
+ pages = "275282",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/EKM96.pdf",
+ paper = "Erli96.pdf"
}
\end{chunk}
\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"
+@InProceedings{Kalt96,
+ author = "Kaltofen, E. and Lobo, A.",
+ title = "On rank properties of {Toeplitz} matrices over finite fields",
+ booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC96",
+ year = "1996",
+ pages = "241249",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_issac.pdf",
+ paper = "Kalt96.pdf"
}
\end{chunk}
\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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",
+@Article{Kalt97,
+ author = "Kaltofen, E.",
+ title = "Teaching Computational Abstract Algebra",
+ journal = "Journal of Symbolic Computation",
+ volume = "23",
+ number = "56",
+ pages = "503515",
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"
+ 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",
+ 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{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",
+\begin{chunk}{axiom.bib}
+@Unpublished{Hitz97,
+ author = "Hitz, M. A. and Kaltofen, E.",
+ title = "The {Kharitonov} theorem and its applications in symbolic
+ mathematical computation",
+ year = "1997",
+ month = "May",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/HiKa97_kharit.pdf",
+ paper = "Hitz97.pdf"
+}
\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"
+@InProceedings{Bern99,
+ author = "Bernardin, L. and Char, B. and Kaltofen, E.",
+ title = "Symbolic Computation in {Java}: an Appraisement",
+ booktitle = "Proc. 1999 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC99",
+ year = "1999",
+ pages = "237244",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/BCK99.pdf",
+ paper = "Bern99.pdf"
}
\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}{axiom.bib}
+@InProceedings{Kalt02,
+ author = "Kaltofen, Erich and McLean, Michael and Norris, Larry",
+ title = "`{Using} {Maple} to Grade {Maple}' Assessment Software from
+ {North Carolina State University}",
+ booktitle = "Proceedings 2002 Maple Workshop",
+ year = "2002",
+ publisher = "Waterloo Maple Inc.",
+ address = "Waterloo, Canada",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/KMN02.pdf",
+ paper = "Kalt02.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Daly 03]{Dal03} Daly, Timothy
``The Axiom Wiki Website''
\verbaxiom.axiomdeveloper.org
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Book{Grab03,
+ editor = "Grabmeier, J. and Kaltofen, E. and Weispfenning, V.",
+ title = "Computer Algebra Handbook",
+ publisher = "SpringerVerlag",
+ year = "2003",
+ note = "637 + xx~pages + CDROM. Includes E. Kaltofen and V. Weispfenning
+ \S1.4 Computer algebra  impact on research, pages 46;
+ E. Kaltofen
+ \S2.2.3 Absolute factorization of polynomials, page 26;
+ E. Kaltofen and B. D. Saunders
+ \S2.3.1 Linear systems, pages 3638;
+ R. M. Corless, E. Kaltofen and S. M. Watt
+ \S2.12.3 Hybrid methods, pages 112125;
+ E. Kaltofen
+ \S4.2.17 FoxBox and other blackbox systems, pages 383385.",
+ isbn = "3540654666",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/symnum.pdf",
+ paper = "Grab03.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt07,
+ author = "Kaltofen, Erich and Li, Bin and Sivaramakrishnan, Kartik and
+ Yang, Zhengfeng and Zhi, Lihong",
+ title = "Lower bounds for approximate factorizations via semidefinite
+ programming (extended abstract)",
+ year = "2007",
+ booktitle =
+ "SNC'07 Proc. 2007 Internat. Workshop on SymbolicNumeric Comput.",
+ crossref = "SNC07",
+ pages = "203204",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KLSYZ07.pdf",
+ paper = "Kalt07.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Daly 09]{Dal09} Daly, Timothy
``The Axiom Literate Documentation''
\verbaxiomdeveloper.org/axiomwebsite/documentation.html
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Article{Borw07,
+ author = "Borwein, Peter and Kaltofen, Erich and Mossinghoff, Michael J.",
+ title = "Irreducible Polynomials and {Barker} Sequences",
+ journal = "{ACM} Communications in Computer Algebra",
+ volume = "162",
+ number = "4",
+ year = "2007",
+ pages = "118121",
+ month = "December",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/BKM07.pdf",
+ paper = "Borw07.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@Article{Kalt10,
+ author = "Kaltofen, Erich and Lavin, Mark",
+ title = "Efficiently Certifying NonInteger Powers",
+ journal = "Computational Complexity",
+ year = "2010",
+ volume = "19",
+ number = "3",
+ month = "September",
+ pages = "355366",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/09/KaLa09.pdf",
+ paper = "Kalt10.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt11,
+ author = "Kaltofen, Erich L. and Nehring, Michael",
+ title = "Supersparse black box rational function interpolation",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'11",
+ crossref = "ISSAC11",
+ month = "June",
+ year = "2011",
+ pages = "177185",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KaNe11.pdf",
+ paper = "Kalt11.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Gren11a,
+ author = "Grenet, Bruno and Kaltofen, Erich L. and Koiran, Pascal
+ and Portier, Natacha",
+ title = "Symmetric Determinantal Representation of Weakly Skew Circuits",
+ booktitle = "Proc. 28th Internat. Symp. on Theoretical Aspects of Computer
+ Science, STACS 2011",
+ crossref = "STACS11",
+ pages = "543554",
+ year = "2011",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/GKKP11.pdf",
+ paper = "Gren11a.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt11a,
+ author = "Kaltofen, Erich L. and Nehring, Michael and Saunders, David B.",
+ title = "QuadraticTime Certificates in Linear Algebra",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'11",
+ crossref = "ISSAC11",
+ month = "June",
+ year = "2011",
+ pages = "171176",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KNS11.pdf",
+ paper = "Kalt11a.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 84a]{Dav84a} Davenport, James H.
``A New Algebra System''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Dav84a.pdf
 keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt11b,
+ author = "Kaltofen, Erich L. and Lee, Wenshin and Yang, Zhengfeng",
+ title = "Fast estimates of {Hankel} matrix condition numbers
+ and numeric sparse interpolation",
+ booktitle = "Proc. 2011 Internat. Workshop on SymbolicNumeric Comput.",
+ month = "June",
+ crossref = "SNC11",
+ year = "2011",
+ pages = "130136",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KLY11.pdf",
+ paper = "Kalt11b.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Guo12,
+ author = "Guo, Feng and Kaltofen, Erich L. and Zhi, Lihong",
+ title = "Certificates of Impossibility of {Hilbert}{Artin} Representations
+ of a Given Degree for Definite Polynomials and Functions",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'12",
+ crossref = "ISSAC12",
+ month = "July",
+ year = "2012",
+ pages = "195202",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/12/GKZ12.pdf",
+ paper = "Guo12.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Come12a,
+ author = "Comer, Matthew T. and Kaltofen, Erich L. and Pernet, Cl{\'e}ment",
+ title = "Sparse Polynomial Interpolation and {Berlekamp}/\allowbreak
+ {Massey} Algorithms That Correct Outlier Errors in Input Values",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'12",
+ crossref = "ISSAC12",
+ month = "July",
+ year = "2012",
+ pages = "138145",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/12/CKP12.pdf",
+ paper = "Come12a.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Boye13,
+ author = "Boyer, Brice and Comer, Matthew T. and Kaltofen, Erich L.",
+ title = "Sparse Polynomial Interpolation by Variable Shift in
+ the Presence of Noise and Outliers in the Evaluations",
+ booktitle =
+ "Proc. Tenth Asian Symposium on Computer Mathematics (ASCM 2012)",
+ year = "2013",
+ month = "October",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/13/BCK13.pdf",
+ paper = "Boye13.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt13b,
+ author = "Kaltofen, Erich and Yang, Zhengfeng",
+ title = "Sparse multivariate function recovery from values with noise and
+ outlier errors",
+ year = "2013",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'13",
+ crossref = "ISSAC13",
+ pages = "219226",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/13/KaYa13.pdf",
+ paper = "Kalt13b.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt13c,
+ author = "Kaltofen, Erich L.",
+ title = "Symbolic Computation and Complexity Theory Transcript of My Talk",
+ booktitle =
+ "Proc. Tenth Asian Symposium on Computer Mathematics (ASCM 2012)",
+ year = "2013",
+ month = "October",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/13/Ka13.pdf",
+ paper = "Kalt13c.pdf"
+}
\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 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 $\overline{Z}[\sqrt{5}][x]$"
+@InProceedings{Kalt14,
+ author = "Kaltofen, Erich L. and Yang, Zhengfeng",
+ title = "Sparse Multivariate Function Recovery With a High Error Rate
+ in Evaluations",
+ year = "2014",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'14",
+ crossref = "ISSAC14",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/KaYa14.pdf",
+ paper = "Kalt14.pdf"
}
\end{chunk}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt14a,
+ author = "Kaltofen, Erich L. and Pernet, Cl{\'e}ment",
+ title = "Sparse Polynomial Interpolation Codes and Their Decoding
+ Beyond Half the Minimal Distance",
+ year = "2014",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'14",
+ crossref = "ISSAC14",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/KaPe14.pdf",
+ paper = "Kalt14a.pdf"
+}
\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",
+\begin{chunk}{axiom.bib}
+@InProceedings{Duma14,
+ author = "Dumas, JeanGuillaume and Kaltofen, Erich L.",
+ title = "Essentially Optimal Interactive Certificates In Linear Algebra",
+ year = "2014",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'14",
+ crossref = "ISSAC14",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/DuKa14.pdf",
+ paper = "Duma14.pdf"
+}
\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",
 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."
+\begin{chunk}{axiom.bib}
+@InProceedings{Boye14,
+ author = "Boyer, Brice B. and Kaltofen, Erich L.",
+ title = "Numerical Linear System Solving With Parametric Entries By
+ Error Correction",
+ year = "2014",
+ booktitle = "SNC'14 Proc. 2014 Int. Workshop on SymbolicNumeric Comput.",
+ crossref = "SNC14",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/BoKa14.pdf",
+ paper = "Boye14.pdf"
+}
\end{chunk}
\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",
+\section{Axiom Citations in the Literature}
\end{chunk}
+\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
+\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[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
+\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",
 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{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
+\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"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davenport 12]{Dav12} Davenport, J.H.
``Computer Algebra''
\verbstaff.bath.ac.uk/masjhd/JHDCA.pdf
+\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[Davenport (b)]{DSTxx} Davenport, J. H.; Siret; Tournier
``Computer Algebra'' \hfill
\verbstaff.bath.ac.uk/masjhd/masternew.pdf
+\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[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)
+\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}{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[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{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
+\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[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
+\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}
+
+\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",
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."
+ 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{chunk}{ignore}
\bibitem[DLMF]{DLMF}.
``Digital Library of Mathematical Functions''
\verbdlmf.nist.gov/software/#T1
+\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{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
+\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{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
+\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 = "
 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."
+ 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[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
+\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 = "
 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.
+ 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.
 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.
+ 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."
+}
 We shall list some assumptions so that we may prove new results so
 that all coercions are unique. This concept is called ``coherence''.
+\end{chunk}
 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."
+\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[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
+\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{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",
 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."
+\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[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
+\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[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
+\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 = "
 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."
+ 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{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
+\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[Duval 94]{Duv94} Duval, Dominique
``Symbolic or algebraic computation?''
Madrid Spain, NAG conference (private copy of paper)
+\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}{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"
+\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}
\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
+\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 = "
 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."

+ 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}
\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
+\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[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
+\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[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
+\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 = "
 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."
+ 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{chunk}{ignore}
\bibitem[Faure 00a]{FDN00a} Faure, Christ\'ele; Davenport, James
``Parameters in Computer Algebra''
+\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}
\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
+\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 = "
 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[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",
+ 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{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
+\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{chunk}{ignore}
\bibitem[Fogus 11]{Fog11} Fogus, Michael
``UnConj''
\verbclojure.com/blog/2011/11/22/unconj.html
+\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}
\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",
+\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}{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",
+\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"
+}
\end{chunk}
\begin{chunk}{ignore}
\bibitem[FSF 14]{FSF14} FSF
``Free Software Directory''
\verbdirectory.fsf.org/wiki/Axiom
+\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",
+ 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[Frisco ]{Fris} Frisco
``Objectives and Results''
\verbwww.nag.co.uk/projects/frisco/frisco/node3.htm
+\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}
\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",
+\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[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
+\bibitem[Burge 91]{Burg91}
+ author = "Burge, W.H.",
+ title = "Scratchpad and the RogersRamanujan identities",
+ year = "1991",
+ pages = "189190",
+ isbn = "0897914376",
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."
+ This note sketches the part played by Scratchpad in obtaining new
+ proofs of Euler's theorem and the RogersRamanujan Identities."
\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",
+@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[Gianni 87]{Gia87} Gianni, Patrizia
``Primary Decomposition of Ideals''
in [Wit87], pp1213
+\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[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
+\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}
\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",

\end{chunk}
+\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
+\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[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
+\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[GomezDiaz 92]{Gom92} G\'omezD'iaz, Teresa
``Quelques applications de l`\'evaluation dynamique''
Ph.D. Thesis L'Universite De Limoges March 1992
+\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[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."
+\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[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",
+\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[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",
+\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[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",
+\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[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
+\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[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
+\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",
 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."
+Proc. Natl. Acad. Sci. USA Vol 86
\end{chunk}
\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",
 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."
+\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{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
+\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{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
+\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{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
+\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{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
+\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 = "
 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."
+ 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{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",
+\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{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",

\end{chunk}
+\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
+@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{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)
+\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}{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}
+@misc{Daly88,
+ author = "Daly, Timothy",
+ title = "Axiom in an Educational Setting, Axiom course slide deck",
+ year = "1988",
+ month = "January",
+ keywords = "axiomref"
+}
\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",
 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."
+\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}
\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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
+\bibitem[Daly 03]{Dal03} Daly, Timothy
+``The Axiom Wiki Website''
+\verbaxiom.axiomdeveloper.org
keywords = "axiomref",
\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
+\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[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)
+\bibitem[Daly 09]{Dal09} Daly, Timothy
+``The Axiom Literate Documentation''
+\verbaxiomdeveloper.org/axiomwebsite/documentation.html
keywords = "axiomref",
\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
+\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{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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
+\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[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
+\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[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
+\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}
\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
+\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[Jenks 74]{Jen74} Jenks, R. D.
``The SCRATCHPAD language''
ACM SIGPLAN Notices, 9(4) pp101111 1974 CODEN SINODQ. ISSN 03621340
+\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[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
+\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}
\begin{chunk}{ignore}
\bibitem[Jenks 79]{Jen79} Jenks, R. D.
``MODLISP: An Introduction''
Proc EUROSAM 79, pp466480, 1979 and IBMRC8073 Jan 1980
+\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[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
+\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[Jenks 81a]{JT81a} Jenks, R.D.; Trager, B.M.
``A Language for Computational Algebra''
SIGPLAN Notices, New York: Association for Computing Machiner, Nov 1981
+\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}{ignore}
\bibitem[Jenks 81b]{JT81b} Jenks, R.D.; Trager, B.M.
``A Language for Computational Algebra''
IBM Research Report RC8930 IBM Yorktown Heights, NY
+\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 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[Jenks 84a]{Jen84a} Jenks, Richard D.
``The new SCRATCHPAD language and system for computer algebra''
In Golden and Hussain [GH84], pp409??
+\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[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
+\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[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
+\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 = "
 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."
+ 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[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)
+\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[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
+\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{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
+\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{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",
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Davenport 12]{Dav12} Davenport, J.H.
+``Computer Algebra''
+\verbstaff.bath.ac.uk/masjhd/JHDCA.pdf
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Jenks 88b]{Jen88b} Jenks, Richard
``The Scratchpad II Computer Algebra System Interactive Environment Users
Guide''
 Spring 1988
+\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[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
+\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}
@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"
+@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[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
+\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[Joswig 03]{JT03} Joswig, Michael; Takayama, Nobuki
``Algebra, geometry, and software systems''
SpringerVerlag ISBN 3540002561 p291
+\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[Joyner 06]{J006} Joyner, David
``OSCAS  Maxima''
SIGSAM Communications in Computer Algebra, 157 2006
\verbsage.math.washington.edu/home/wdj/sigsam/oscascca1.pdf
+\bibitem[DLMF]{DLMF}.
+``Digital Library of Mathematical Functions''
+\verbdlmf.nist.gov/software/#T1
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Joyner 14]{JO14} Joyner, David
``Links to some open source mathematical programs''
\verbwww.opensourcemath.org/opensource_math.html
+\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}
\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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
+\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 = "
 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."

+ 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[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
+\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''.
+
+ 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.
+
+ 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.
+
+ We shall list some assumptions so that we may prove new results so
+ that all coercions are unique. This concept is called ``coherence''.
+
+ 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[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
+\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[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
+\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 = "
 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."
+ 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[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
+\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[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
+\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[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
+\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[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
+\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[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",
+\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{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{E} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Lahey 08]{Lah08} Lahey, Tim
``Sage Integration Testing''
\verbgithub.com/tjl/sage_int_testing Dec. 2008
+\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[Lambe 89]{Lam89} Lambe, L. A.
``Scratchpad II as a tool for mathematical research''
Notices of the AMS, February 1928 pp143147
+\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[Lambe 91]{Lam91} Lambe, L. A.
``Resolutions via homological perturbation''
Journal of Symbolic Computation 12(1) pp7187 July 1991
CODEN JSYCEH ISSN 07477171
+\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[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
+\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 = "
 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
 \[1+t+\ldots+t^i=\sum_j{t^j}\]
+ 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.
 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."
+ 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[Lambe 93]{Lam93} Lambe, Larry
``On Using Axiom to Generate Code''
(preprint) 1993
+\bibitem[Faure 00a]{FDN00a} Faure, Christ\'ele; Davenport, James
+``Parameters in Computer Algebra''
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
+\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 = "
 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}
+ 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}{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",
+ 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[LeBlanc 91]{LeB91} LeBlanc, S.E.
``The use of MathCAD and Theorist in the ChE classroom''
In Anonymous [Ano91], pp287299 (vol. 1) 2 vols.
+\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{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
+\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{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
+\bibitem[Fogus 11]{Fog11} Fogus, Michael
+``UnConj''
+\verbclojure.com/blog/2011/11/22/unconj.html
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
+\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[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
+\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{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
+\bibitem[FSF 14]{FSF14} FSF
+``Free Software Directory''
+\verbdirectory.fsf.org/wiki/Axiom
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
+\bibitem[Frisco ]{Fris} Frisco
+``Objectives and Results''
+\verbwww.nag.co.uk/projects/frisco/frisco/node3.htm
keywords = "axiomref",
 abstract = "
 This paper proposes a nonintrusive automatic parallelization
 framework for typeful and propertyaware computer algebra systems."
\end{chunk}
+\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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
+\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",
 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[Linton 93]{Lin93} Linton, Steve
``Vector Enumeration Programs, version 3.04''
\verbwww.cs.standrews.ac.uk/~sal/nme/nme_toc.html#SEC1
+\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[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",
+\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[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
+\bibitem[Gianni 87]{Gia87} Gianni, Patrizia
+``Primary Decomposition of Ideals''
+in [Wit87], pp1213
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
+\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[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
+\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}
\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)
+\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[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
+\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{chunk}{ignore}
\bibitem[McJones 11]{McJ11} McJones, Paul
``Software Presentation Group  Common Lisp family''
\verbwww.softwarepreservation.org/projects/LISP/common_lisp_family
+\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[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
+\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[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
+\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[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
+\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[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
+\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",
 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[Monagan 87]{Mon87} Monagan, Michael B.
``Support for Data Structures in Scratchpad II''
in [Wit87], pp1718
+\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[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
+\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[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
+\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[Moses 71]{Mos71} Moses, Joel
``Algebraic Simplification: A Guide for the Perplexed''
CACM August 1971 Vol 14 No. 8 pp527537
+\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[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
+\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",
 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[Naylor]{NPxx} Naylor, William; Padget, Julian
``From Untyped to Polymorphically Typed Objects in Mathematical Web
Services''
%\verbaxiomdeveloper.org/axiomwebsite/papers/NPxx.pdf
+\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",
 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[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
+\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 = "
 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."
+ 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[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
+\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",
 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[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
+\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[Norman 75a]{Nor75a} Norman, A.C.
``The SCRATCHPAD Power Series Package''
IBM T.J. Watson Research RC4998
+\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}
\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
+\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}
\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.
+\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[OpenMath]{OpenMa}.
``OpenMath Technical Overview''
\verbwww.openmath.org/overview/technical.html
+\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{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
+\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[Petitot 90]{Pet90} Petitot, Michel
``Types r\'ecursifs en scratchpad, application aux polyn\^omes non
commutatifs''
LIFL, 1990
+\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[Petitot 93]{Pet93} Petitot, M.
``Experience with Axiom''
In Jacob et al. [JOS93], page 240
+\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[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
+\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[Pinch 93]{Pin93} Pinch, R.G.E.
``Some Primality Testing Algorithms''
Devlin, Keith (ed.)
Computers and Mathematics November 1993, Vol 40, Number 9 pp12031210
+\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[Poll (b)]{Polxx} Poll, Erik
``The type system of Axiom''
%\verbaxiomdeveloper.org/axiomwebsite/papers/Polxx.pdf
+\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[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
+\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}
\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Rainer 14]{Rain14} Joswig, Rainer
``2014: 30+ Years Common Lisp the Language''
\verblispm.de/30ycltl
+\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[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
+\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[Rioboo 03]{Riob03} Rioboo, Renaud
``Towards Faster Real Algebraic Numbers''
J. of Symbolic Computation 36 pp 513533 (2003)
%\verbaxiomdeveloper.org/axiomwebsite/papers/Riob03.pdf
+\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",
 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[Robidoux 93]{Rob93} Robidoux, Nicolas
``Does Axiom Solve Systems of O.D.E's Like Mathematica?''
July 1993
%\verbaxiomdeveloper.org/axiomwebsite/papers/Rob93.pdf
+\bibitem[Jenks 79]{Jen79} Jenks, R. D.
+``MODLISP: An Introduction''
+Proc EUROSAM 79, pp466480, 1979 and IBMRC8073 Jan 1980
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[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
+\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",
 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[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
+\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}
\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Sage 14]{Sage14} Stein, William
``Sage''
\verbwww.sagemath.org/doc/reference/interfaces/sage/interfaces/axiom.html
+\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[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[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[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[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}{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"
}
+\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",
+ 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[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
+\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[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[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[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[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[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[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[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[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",
 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}{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{chunk}{ignore}
\bibitem[Seiler 94a]{Sei94a} Seiler, W.M.
``Completion to involution in AXIOM''
in Calmet [Cal94] pp103104
+\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{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[Joswig 03]{JT03} Joswig, Michael; Takayama, Nobuki
+``Algebra, geometry, and software systems''
+SpringerVerlag ISBN 3540002561 p291
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}
\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
+\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",
 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[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[Joyner 14]{JO14} Joyner, David
+``Links to some open source mathematical programs''
+\verbwww.opensourcemath.org/opensource_math.html
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}
+\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[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 = "
 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."
+ 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{chunk}{ignore}
\bibitem[Seiler (a)]{Seixx} Seiler, W.M.
``DETools: A Library for Differential Equations''
\verbiakswww.ira.uka.de/iakscalmet/werner/werner.html
+\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}{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
+\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[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
+\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}{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[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",
 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[Sit 06]{Sit06} Sit, Emil
``Tools for Repeatable Research''
\verbwww.emilsit.net/blog/archives/toolsforrepeatableresearch
+\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[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
+\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[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[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",
 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}
+\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[Lahey 08]{Lah08} Lahey, Tim
+``Sage Integration Testing''
+\verbgithub.com/tjl/sage_int_testing Dec. 2008
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[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[Steele]{Steele} Steele, Guy L.; Gabriel, Richard P.
``The Evolution of Lisp''
\verbwww.dreamsongs.com/Files/HOPL2Uncut.pdf
+\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[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[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[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[Lambe 93]{Lam93} Lambe, Larry
+``On Using Axiom to Generate Code''
+(preprint) 1993
keywords = "axiomref",
\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[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}{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[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",
 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}
\begin{chunk}{ignore}
\bibitem[Sutor 88]{Su88} Sutor, Robert S.
``A guide to programming in the scratchpad 2 interpreter''
IBM Manual, March 1988
+\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}
\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
+\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",
 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[Thompson (a)]{TTxx} Thompson, Simon; Timochouk, Leonid
``The Aldor\\ language''
%\verbaxiomdeveloper.org/axiomwebsite/papers/TTxx.pdf
+\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",
 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[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[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}
\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
+\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}{axiom.bib}
@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",
+\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",
 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.
+\end{chunk}
 The Puiseux expansions generally introduce algebraic extensions. These
 extensions will not appear in the resulting integral basis."
}
+\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",
+ abstract = "
+ This paper proposes a nonintrusive automatic parallelization
+ framework for typeful and propertyaware computer algebra systems."
\end{chunk}
\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",
+\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",
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."
}
+ 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[Vasconcelos 99]{Vas99} Vasconcelos, Wolmer
``Computational Methods in Commutative Algebra and Algebraic Geometry''
Springer, Algorithms and Computation in Mathematics, Vol 2 1999
ISBN 3540213112
+\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}
\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
+\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[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[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[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
+\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[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[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[Watt 85]{Wat85} Watt, Stephen
``Bounded Parallelism in Computer Algebra''
PhD Thesis, University of Waterloo
\verbwww.csd.uwo.ca/~watt/pub/reprints/1985smwphd.pdf
+\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[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[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[Watt 87]{Wat87} Watt, Stephen
``Domains and Subdomains in Scratchpad II''
in [Wit87], pp35
+\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[Watt 87a]{WB87} Watt, Stephen M.; Burge, William H.
``Mapping as First Class Objects''
in [Wit87], pp1317
+\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[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[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[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[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[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
+\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[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
+\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[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
+\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[Watt 94c]{Wat94c} Watt, Stephen M.
``A\# Language Reference Version 0.35''
IBM Research Division Technical Report RC19530 May 1994
+\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[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[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[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[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[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
+\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 = "
 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."
+ 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[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
+\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 = "
 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."
+ 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[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[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 = "
 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."
+ 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[Weber 94]{Web94} Weber, Andreas
``Algorithms for Type Inference with Coercions''
ISSAC 94 ACM 0897916387/94/0007
%\verbaxiomdeveloper.org/axiomwebsite/papers/Web94.pdf
+\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",
 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[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
+\bibitem[Norman 75a]{Nor75a} Norman, A.C.
+``The SCRATCHPAD Power Series Package''
+IBM T.J. Watson Research RC4998
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}
+\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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
+\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",
 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[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[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",
 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[WeiJiang 12]{WJ12} WeiJiang
``Top free algebra System''
\verbweijiang.com/it/software/topfreealgebrasystembyemathematicabyemaple
+\bibitem[OpenMath]{OpenMa}.
+``OpenMath Technical Overview''
+\verbwww.openmath.org/overview/technical.html
keywords = "axiomref",
\end{chunk}
+\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
\bibitem[Wester 99]{Wes99} Wester, Michael J.
``Computer Algebra Systems''
John Wiley and Sons 1999 ISBN 0471983535
+\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",
\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[Petitot 90]{Pet90} Petitot, Michel
+``Types r\'ecursifs en scratchpad, application aux polyn\^omes non
+commutatifs''
+LIFL, 1990
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wityak 87]{Wit87} Wityak, Sandra
``Scratchpad II Newsletter''
Volume 2, Number 1, Nov 1987
+\bibitem[Petitot 93]{Pet93} Petitot, M.
+``Experience with Axiom''
+In Jacob et al. [JOS93], page 240
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[WWW1]{WWW1}.
Software Preservation Group
\verbwww.softwarepresentation.org/projects/LISP/common_lisp_family
+\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}
\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Yap 00]{Yap00} Yap, Chee Keng
``Fundamental Problems of Algorithmic Algebra''
Oxford University Press (2000) ISBN0195125169
+\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[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[Poll (b)]{Polxx} Poll, Erik
+``The type system of Axiom''
+%\verbaxiomdeveloper.org/axiomwebsite/papers/Polxx.pdf
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[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{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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[Rainer 14]{Rain14} Joswig, Rainer
+``2014: 30+ Years Common Lisp the Language''
+\verblispm.de/30ycltl
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Zip92]{Zip92} Zippel, Richard
``Algebraic Computation''
(unpublished) Cornell University Ithaca, NY Sept 1992
+\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}{ignore}
\bibitem[Zwi92]{Zwi92} Zwillinger, Daniel
``Handbook of Integration''
Jones and Bartlett, 1992, ISBN 0867202939
+\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}
\section{Axiom Citations of External Sources}
\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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",
+ 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.
\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"
}
+ 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}{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[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[Abramowitz 64]{AS64} Abramowitz, Milton; Stegun, Irene A.
``Handbook of Mathematical Functions''
(1964) Dover Publications, NY ISBN 0486612724
+\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[Abramowitz 68]{AS68} Abramowitz M; Stegun I A
``Handbook of Mathematical Functions''
Dover Publications. (1968)
+\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[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[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}{axiom.bib}
@book{Altm05,
 author = "Altmann, Simon L.",
 title = "Rotations, Quaternions, and Double Groups",
 publisher = "Dover Publications, Inc.",
 year = "2005",
 isbn = "0486445186"
+@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[Ames 77]{Ames77} Ames W F
``Nonlinear Partial Differential Equations in Engineering''
Academic Press (2nd Edition). (1977)
+\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}{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[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[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[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}
\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[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[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[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 = "
 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."
+ 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}{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."
}
+\begin{chunk}{ignore}
+\bibitem[Seiler 94a]{Sei94a} Seiler, W.M.
+``Completion to involution in AXIOM''
+in Calmet [Cal94] pp103104
+ keywords = "axiomref",
\end{chunk}
\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[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 = "
 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."
+ 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{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[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[Baker 96]{BGM96} Baker, George A.; GravesMorris, Peter
``Pade Approximants''
Cambridge University Press, March 1996 ISBN 9870521450072
+\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[Baker 10]{Ba10} Baker, Martin
``3D World Simulation''
\verbwww.euclideanspace.com

\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"
}
+\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[Banks 68]{BK68} Banks D O; Kurowski I
``Computation of Eigenvalues of Singular SturmLiouville Systems''
Math. Computing. 22 304310. (1968)
+\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[Bard 74]{Bard74} Bard Y
``Nonlinear Parameter Estimation''
Academic Press. 1974
+\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}
\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[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[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[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[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[Sit 06]{Sit06} Sit, Emil
+``Tools for Repeatable Research''
+\verbwww.emilsit.net/blog/archives/toolsforrepeatableresearch
+ keywords = "axiomref",
\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[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}{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",
+\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
+ keywords = "axiomref",
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."
}
+ 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[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[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}{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[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[Birkhoff 62]{BR62} Birkhoff, G; Rota, G C
``Ordinary Differential Equations''
Ginn \& Co., Boston and New York. (1962)
+\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[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[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[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[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[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[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[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.
+\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}
\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[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}{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

\end{chunk}
+\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Brigham 73]{Bri73} Brigham E O
``The Fast Fourier Transform''
PrenticeHall. (1973)
+\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[Brillhart 69]{Bri69} Brillhart, John
``On the Euler and Bernoulli polynomials''
J. Reine Angew. Math., v. 234, (1969), pp. 4564
+\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[Brillhart 90]{Bri90} Brillhart, John
``Note on Irreducibility Testing''
Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 13791381
+\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}
\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)

\end{chunk}
+\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
 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."
+\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}
@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",
+@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 = "
 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."
+ 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}{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"
+@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[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.
+\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",
 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.
+\end{chunk}
 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."
+\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
+ keywords = "axiomref",
\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
 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."
+\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",
\end{chunk}
\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",
 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."
}
+\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}
\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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[Carlson 65]{Car65} Carlson B C
``On Computing Elliptic Integrals and Functions''
J Math Phys. 44 3651. (1965)
+\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}{ignore}
\bibitem[Carlson 77a]{Car77a} Carlson B C
``Elliptic Integrals of the First Kind''
SIAM J Math Anal. 8 231242. (1977)
+\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[Carlson 77b]{Car77b} Carlson B C
``Special Functions of Applied Mathematics''
Academic Press. (1977)
+\bibitem[Watt 87]{Wat87} Watt, Stephen
+``Domains and Subdomains in Scratchpad II''
+in [Wit87], pp35
+ 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[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[Carlson 88]{Car88} Carlson B C,
``A Table of Elliptic Integrals of the Third Kind''
Math. Comput. 51 267280. (1988)
+\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[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[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}{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
 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."
+\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}{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[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}{ignore}
\bibitem[Clausen 89]{Cla89} Clausen, M.; Fortenbacher, A.
``Efficient Solution of Linear Diophantine Equations''
JSC (1989) 8, 201216
+\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[Clenshaw 55]{Cle55} Clenshaw C W,
``A Note on the Summation of Chebyshev Series''
Math. Tables Aids Comput. 9 118120. (1955)
+\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}{ignore}
\bibitem[Clenshaw 60]{Cle60} Clenshaw C W
``Curve Fitting with a Digital Computer''
Comput. J. 2 170173. (1960)
+\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}{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[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}
\begin{chunk}{ignore}
\bibitem[Clenshaw 62]{Cle62} Clenshaw C W
``Mathematical Tables. Chebyshev Series for Mathematical Functions''
HMSO. (1962)
+\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[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[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[Conway 87]{CCNPW87} Conway, J.; Curtis, R.; Norton, S.; Parker, R.;
Wilson, R.
``Atlas of Finite Groups''
Oxford, Clarendon Press, 1987

\end{chunk}
+\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.
\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
+ 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[Cox 72]{Cox72} Cox M G
``The Numerical Evaluation of Bsplines''
J. Inst. Math. Appl. 10 134149. (1972)
+\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}{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[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[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[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[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[WeiJiang 12]{WJ12} WeiJiang
+``Top free algebra System''
+\verbweijiang.com/it/software/topfreealgebrasystembyemathematicabyemaple
+ keywords = "axiomref",
\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[Wester 99]{Wes99} Wester, Michael J.
+``Computer Algebra Systems''
+John Wiley and Sons 1999 ISBN 0471983535
+ 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[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[Cox 78]{Cox78} Cox M G
``The Numerical Evaluation of a Spline from its Bspline Representation''
J. Inst. Math. Appl. 21 135143. (1978)
+\bibitem[Wityak 87]{Wit87} Wityak, Sandra
+``Scratchpad II Newsletter''
+Volume 2, Number 1, Nov 1987
+ 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[WWW1]{WWW1}.
+Software Preservation Group
+\verbwww.softwarepresentation.org/projects/LISP/common_lisp_family
+ keywords = "axiomref",
\end{chunk}
\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Dahlquist 74]{DB74} Dahlquist G; Bjork A
``Numerical Methods''
Prentice Hall. (1974)
+\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[Dalmas 98]{DA98} Dalmas, Stephane; Arsac, Olivier
``The INRIA OpenMath Library''
Projet SAFIR, INRIA Sophia Antipolis Nov 25, 1998
+\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[Dantzig 63]{Dan63} Dantzig G B
``Linear Programming and Extensions''
Princeton University Press. (1963)
+\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[Davenport]{Dav} Davenport, James
``On Brillhart Irreducibility.''
To appear.
+\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[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[Zip92]{Zip92} Zippel, Richard
+``Algebraic Computation''
+(unpublished) Cornell University Ithaca, NY Sept 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Davis 67]{DR67} Davis P J; Rabinowitz P
``Numerical Integration''
Blaisdell Publishing Company. 3352. (1967)
+\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[Davis 75]{DR75} Davis P J; Rabinowitz P
``Methods of Numerical Integration''
Academic Press. (1975)
+\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}{ignore}
\bibitem[DeBoor 72]{DeB72} De Boor C
``On Calculating with Bsplines''
J. Approx. Theory. 6 5062. (1972)
+\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"
+}
\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[Abramowitz 64]{AS64} Abramowitz, Milton; Stegun, Irene A.
+``Handbook of Mathematical Functions''
+(1964) Dover Publications, NY ISBN 0486612724
\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[Abramowitz 68]{AS68} Abramowitz M; Stegun I A
+``Handbook of Mathematical Functions''
+Dover Publications. (1968)
\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
+\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[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[Ames 77]{Ames77} Ames W F
+``Nonlinear Partial Differential Equations in Engineering''
+Academic Press (2nd Edition). (1977)
\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[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[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[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}
\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[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[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[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[Dongarra 79]{DMBS79} Dongarra J J; Moler C B; Bunch J R;
Stewart G W
``LINPACK Users' Guide''
SIAM, Philadelphia. (1979)
+\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[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[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[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[Bailey 66]{Bai66} Bailey P B
+``SturmLiouville Eigenvalues via a Phase Function''
+SIAM J. Appl. Math . 14 242249. (1966)
\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[Baker 96]{BGM96} Baker, George A.; GravesMorris, Peter
+``Pade Approximants''
+Cambridge University Press, March 1996 ISBN 9870521450072
\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[Baker 10]{Ba10} Baker, Martin
+``3D World Simulation''
+\verbwww.euclideanspace.com
\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
+\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[Ducos 00]{Duc00} Ducos, Lionel
``Optimizations of the subresultant algorithm''
Journal of Pure and Applied Algebra V145 No 2 Jan 2000 pp149163
+\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[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[Bard 74]{Bard74} Bard Y
+``Nonlinear Parameter Estimation''
+Academic Press. 1974
\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
 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''."
+\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[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."
+\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[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."
+\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[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."
+\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}{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
+\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",
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."
+ 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}
\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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
 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?"
+\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}{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[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}{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[Birkhoff 62]{BR62} Birkhoff, G; Rota, G C
+``Ordinary Differential Equations''
+Ginn \& Co., Boston and New York. (1962)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Fletcher 81]{Fle81} Fletcher R
``Practical Methods of Optimization''
Vol 2. Constrained Optimization. Wiley. (1981)
+\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}{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[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[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[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[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."
+\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[Fox 68]{Fox68} Fox L.; Parker I B.
``Chebyshev Polynomials in Numerical Analysis''
Oxford University Press. (1968)
+\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[Franke 80]{FN80} Franke R.; Nielson G
``Smooth Interpolation of Large Sets of Scattered Data''
Internat. J. Num. Methods Engrg. 15 16911704. (1980)
+\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[Fritsch 82]{Fri82} Fritsch F N
``PCHIP Final Specifications''
Report UCID30194. Lawrence Livermore National Laboratory. (1982)
+\bibitem[Brigham 73]{Bri73} Brigham E O
+``The Fast Fourier Transform''
+PrenticeHall. (1973)
\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[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[Froberg 65]{Fro65} Froberg C E.
``Introduction to Numerical Analysis''
AddisonWesley. 181187. (1965)
+\bibitem[Brillhart 90]{Bri90} Brillhart, John
+``Note on Irreducibility Testing''
+Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 13791381
\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[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[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[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 = "
 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."
+ 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}{ignore}
\bibitem[Gathen 90]{Gat90} Gathen, Joachim von zur
``Functional Decomposition Polynomials: the Tame Case''
Journal of Symbolic Computation (1990) 9, 281299
+\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",
+ 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}{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"
+@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[Gautschi 79a]{Gau79a} Gautschi W.
``A Computational Procedure for Incomplete Gamma Functions''
ACM Trans. Math. Softw. 5 466481. (1979)
+\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
+ 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[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}
+@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}
\begin{chunk}{ignore}
\bibitem[Gautschi 79b]{Gau79b} Gautschi W.
``Algorithm 542: Incomplete Gamma Functions''
ACM Trans. Math. Softw. 5 482489. (1979)

\end{chunk}
+\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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[Carlson 65]{Car65} Carlson B C
+``On Computing Elliptic Integrals and Functions''
+J Math Phys. 44 3651. (1965)
\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[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[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[Carlson 77b]{Car77b} Carlson B C
+``Special Functions of Applied Mathematics''
+Academic Press. (1977)
\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[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[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[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[Gill 72]{GM72} Gill P E.; Miller G F.
``An Algorithm for the Integration of Unequally Spaced Data''
Comput. J. 15 8083. (1972)
+\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[Gill 74b]{GM74b} Gill P E.; Murray W. (eds)
``Numerical Methods for Constrained Optimization''
Academic Press. (1974)
+\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[Gill 76a]{GM76a} Gill P E.; Murray W.
``Minimization subject to bounds on the variables''
Report NAC 72. National Physical Laboratory. (1976)
+\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[Gill 76b]{GM76b} Gill P E.; Murray W.
``Algorithms for the Solution of the Nonlinear Leastsquares Problem''
NAC 71 National Physical Laboratory. (1976)
+\bibitem[Clausen 89]{Cla89} Clausen, M.; Fortenbacher, A.
+``Efficient Solution of Linear Diophantine Equations''
+JSC (1989) 8, 201216
\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[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[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[Clenshaw 60]{Cle60} Clenshaw C W
+``Curve Fitting with a Digital Computer''
+Comput. J. 2 170173. (1960)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Gill 81]{GMW81} Gill P E.; Murray W.; Wright M H.
``Practical Optimization''
Academic Press. 1981
+\bibitem[Clenshaw 62]{Cle62} Clenshaw C W
+``Mathematical Tables. Chebyshev Series for Mathematical Functions''
+HMSO. (1962)
\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[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[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[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[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[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[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[Cox 72]{Cox72} Cox M G
+``The Numerical Evaluation of Bsplines''
+J. Inst. Math. Appl. 10 134149. (1972)
\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[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[Gladwell 79]{Gla79} Gladwell I
``Initial Value Routines in the NAG Library''
ACM Trans Math Softw. 5 386400. (1979)
+\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[Gladwell 80]{GS80} Gladwell I.; Sayers D K
``Computational Techniques for Ordinary Differential Equations''
Academic Press. 1980
+\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[Gladwell 86]{Gla86} Gladwell I
``Vectorisation of one dimensional quadrature codes''
Techincal Report. TR7/86 NAG. (1986)
+\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[Gladwell 87]{Gla87} Gladwell I
``The NAG Library Boundary Value Codes''
Numerical Analysis Report. 134 Manchester University. (1987)
+\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}
\begin{chunk}{ignore}
\bibitem[Goedel 40]{God40} Goedel
``The consistency of the continuum hypothesis''
Ann. Math. Studies, Princeton Univ. Press, 1940
+\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[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[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[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[Dahlquist 74]{DB74} Dahlquist G; Bjork A
+``Numerical Methods''
+Prentice Hall. (1974)
\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[Dalmas 98]{DA98} Dalmas, Stephane; Arsac, Olivier
+``The INRIA OpenMath Library''
+Projet SAFIR, INRIA Sophia Antipolis Nov 25, 1998
\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[Dantzig 63]{Dan63} Dantzig G B
+``Linear Programming and Extensions''
+Princeton University Press. (1963)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Grabmeier]{Grab} Grabmeier, J.
``On Plesken's root finding algorithm''
in preparation
+\bibitem[Davenport]{Dav} Davenport, James
+``On Brillhart Irreducibility.''
+To appear.
\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[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[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[Davis 67]{DR67} Davis P J; Rabinowitz P
+``Numerical Integration''
+Blaisdell Publishing Company. 3352. (1967)
\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[Davis 75]{DR75} Davis P J; Rabinowitz P
+``Methods of Numerical Integration''
+Academic Press. (1975)
\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
 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
 \[e^x[exp(1/x+e^{x})exp(1/x)],\quad{}x \leftarrow \infty\]
+\bibitem[DeBoor 72]{DeB72} De Boor C
+``On Calculating with Bsplines''
+J. Approx. Theory. 6 5062. (1972)
 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.
+\end{chunk}
 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."
+\begin{chunk}{ignore}
+\bibitem[De Doncker 78]{DeD78} De Doncker E,
+``An Adaptive Extrapolation Algorithm for Automatic Integration''
+Signum Newsletter. 13 (2) 1218. (1978)
\end{chunk}
\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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[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}{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[Dennis 77]{DM77} Dennis J E Jr; More J J
+``QuasiNewton Methods, Motivation and Theory''
+SIAM Review. 19 4689. 1977
\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[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[Hall 76]{HW76} Hall G.; Watt J M. (eds),
``Modern Numerical Methods for Ordinary Differential Equations''
Clarendon Press. (1976)
+\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}{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[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[Hammarling 85]{Ham85} Hammarling S.
`` The Singular Value Decomposition in Multivariate Statistics''
ACM Signum Newsletter. 20, 3 225. (1985)
+\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{chunk}{ignore}
\bibitem[Hammersley 67]{HH67} Hammersley J M; Handscomb D C.
``MonteCarlo Methods''
Methuen. (1967)
+\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}{axiom.bib}
@misc{Hath1896,
 author = "Hathway, Arthur S.",
 title = "A Primer Of Quaternions",
 year = "1896"
}
+\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)
\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[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[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[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}
\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[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[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[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[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[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[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[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[Higham 02]{Hig02} Higham, Nicholas J.
``Accuracy and stability of numerical algorithms''
SIAM Philadelphia, PA ISBN 0898715210 (2002)
+\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}{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
+\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}
\begin{chunk}{ignore}
\bibitem[Householder 70]{Hou70} Householder A S.
``The Numerical Treatment of a Single Nonlinear Equation''
McGrawHill. (1970)
+\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}
\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[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[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[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}
\subsection{I} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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[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{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[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 = "
 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 $GF(q^m)$
 $(q=2^n)$."
+ 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[Iyanaga 77]{Iya77} Iyanaga, Shokichi; Iyanaga, Yukiyosi Kawada
``Encyclopedic Dictionary of Mathematics''
1977
+\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}
\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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[Jacobson 68]{Jac68} Jacobson, N.
``Structure and Representations of Jordan Algebras''
AMS, Colloquium Publications Volume 39
+\bibitem[Fletcher 81]{Fle81} Fletcher R
+``Practical Methods of Optimization''
+Vol 2. Constrained Optimization. Wiley. (1981)
\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
+\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"
+}
\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[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[Jeffrey 04]{Je04} Jeffrey, Alan
``Handbook of Mathematical Formulas and Integrals''
Third Edition, Elsevier Academic Press ISBN 0123822564
+\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}{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[Fox 68]{Fox68} Fox L.; Parker I B.
+``Chebyshev Polynomials in Numerical Analysis''
+Oxford University Press. (1968)
\end{chunk}
\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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[Kalkbrener 91]{Kal91} Kalkbrener, M.
``Three contributions to elimination theory''
Ph. D. Thesis, University of Linz, Austria, 1991
+\bibitem[Fritsch 82]{Fri82} Fritsch F N
+``PCHIP Final Specifications''
+Report UCID30194. Lawrence Livermore National Laboratory. (1982)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Kalkbrener 98]{Kal98} Kalkbrener, M.
``Algorithmic properties of polynomial rings''
Journal of Symbolic Computation 1998
+\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[Kantor 89]{Kan89} Kantor,I.L.; Solodovnikov, A.S.
``Hypercomplex Numbers''
Springer Verlag Heidelberg, 1989, ISBN 0387969802
+\bibitem[Froberg 65]{Fro65} Froberg C E.
+``Introduction to Numerical Analysis''
+AddisonWesley. 181187. (1965)
\end{chunk}
+\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[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[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[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}{ignore}
\bibitem[Knuth 84]{Knu84} Knuth, Donald
{\it The \TeX{}book}.
Reading, Massachusetts, AddisonWesley Publishing Company, Inc.,
1984. ISBN 0201134489
+\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}
@book{Knut92,
 author = "Knuth, Donald E.",
 title = "Literate Programming",
 publisher = "Center for the Study of Language and Information, Stanford CA",
 year = "1992",
 isbn = "0937073814"
}
+@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[Knu98]{Knu98} Donald Knuth
``The Art of Computer Programming'' Vol. 3
(Sorting and Searching)
AddisonWesley 1998
+\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[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[Gautschi 79b]{Gau79b} Gautschi W.
+``Algorithm 542: Incomplete Gamma Functions''
+ACM Trans. Math. Softw. 5 482489. (1979)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Kolchin 73]{Kol73} Kolchin, E.R.
``Differential Algebra and Algebraic Groups''
(Academic Press, 1973).
+\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[Koutschan 10]{Kou10} Koutschan, Christoph
``Axiom / FriCAS''
\verbwww.risc.jku.at/education/courses/ws2010/cas/axiom.pdf
+\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[Kozen 86]{KL86} Kozen, Dexter; Landau, Susan
``Polynomial Decomposition Algorithms''
Journal of Symbolic Computation (1989) 7, 445456
+\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}
\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\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"
}
+\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)
\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[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[Lawson 77]{Law77} Lawson C L.
``Software for C Surface Interpolation''
Mathematical Software III. (ed J R Rice) Academic Press. 161194. (1977)
+\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[Lawson 74]{LH74} Lawson C L.; Hanson R J.
``Solving Leastsquares Problems''
PrenticeHall. (1974)
+\bibitem[Gill 74b]{GM74b} Gill P E.; Murray W. (eds)
+``Numerical Methods for Constrained Optimization''
+Academic Press. (1974)
\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"
}
+\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)
\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)
+\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[Lazard 91]{Laz91} Lazard, D.
``A new method for solving algebraic systems of positive dimension''
Discr. App. Math. 33:147160,1991
+\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}
\begin{chunk}{ignore}
\bibitem[Lazard92]{Laz92} Lazard, D.
``Solving Zerodimensional Algebraic Systems''
Journal of Symbolic Computation, 1992, 13, 117131
+\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{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."
}
+\begin{chunk}{ignore}
+\bibitem[Gill 81]{GMW81} Gill P E.; Murray W.; Wright M H.
+``Practical Optimization''
+Academic Press. 1981
\end{chunk}
\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[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{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[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{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[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
\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[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{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[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{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[Gladwell 79]{Gla79} Gladwell I
+``Initial Value Routines in the NAG Library''
+ACM Trans Math Softw. 5 386400. (1979)
\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[Gladwell 80]{GS80} Gladwell I.; Sayers D K
+``Computational Techniques for Ordinary Differential Equations''
+Academic Press. 1980
\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[Gladwell 86]{Gla86} Gladwell I
+``Vectorisation of one dimensional quadrature codes''
+Techincal Report. TR7/86 NAG. (1986)
\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[Gladwell 87]{Gla87} Gladwell I
+``The NAG Library Boundary Value Codes''
+Numerical Analysis Report. 134 Manchester University. (1987)
\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[Goedel 40]{God40} Goedel
+``The consistency of the continuum hypothesis''
+Ann. Math. Studies, Princeton Univ. Press, 1940
\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[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
\end{chunk}
\begin{chunk}{ignore}
\bibitem[LTU10]{LTU10}.
``Lambda the Ultimate''
\verblambdatheultimate.org/node/3663#comment62440
+\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}{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[Golub 89]{GL89} Golub, Gene H.; Van Loan, Charles F.
+``Matrix Computations''
+Johns Hopkins University Press ISBN 0801837723 (1989)
\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[Golub 96]{GL96} Golub, Gene H.; Van Loan, Charles F.
+``Matrix Computations''
+Johns Hopkins University Press ISBN 9780801854149 (1996)
\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[Grabmeier]{Grab} Grabmeier, J.
+``On Plesken's root finding algorithm''
+in preparation
\end{chunk}
\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Mac Lane 79]{MB79} Mac Lane, Saunders; Birkhoff, Garret
``Algebra''
AMS Chelsea Publishing ISBN 0821816462
+\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{chunk}{ignore}
\bibitem[Malcolm 72]{Mal72} Malcolm M. A.
``Algorithms to reveal properties of floatingpoint arithmetic''
Comms. of the ACM, 15, 949951. (1972)
+\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{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[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{chunk}{ignore}
\bibitem[Marden 66]{Mar66} Marden M.
``Geometry of Polynomials''
Mathematical Surveys. 3 Am. Math. Soc., Providence, RI. (1966)
+\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{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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"
+@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{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.
+\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{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.
+\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[Maza 98]{Maz98} Maza, M. Moreno
``A new algorithm for computing triangular
decomposition of algebraic varieties''
 NAG Tech. Rep. 4/98.
+\bibitem[Hall 76]{HW76} Hall G.; Watt J M. (eds),
+``Modern Numerical Methods for Ordinary Differential Equations''
+Clarendon Press. (1976)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Mignotte 82]{Mig82} Mignotte, Maurice
``Some Useful Bounds''
Computing, Suppl. 4, 259263 (1982), SpringerVerlag
+\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
\end{chunk}
\begin{chunk}{ignore}
\bibitem[McCarthy 83]{McC83} McCarthy G J.
``Investigation into the Multigrid Code MGD1''
Report AERER 10889. Harwell. (1983)
+\bibitem[Hammarling 85]{Ham85} Hammarling S.
+`` The Singular Value Decomposition in Multivariate Statistics''
+ACM Signum Newsletter. 20, 3 225. (1985)
\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[Hammersley 67]{HH67} Hammersley J M; Handscomb D C.
+``MonteCarlo Methods''
+Methuen. (1967)
\end{chunk}
\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
+\begin{chunk}{axiom.bib}
+@misc{Hath1896,
+ author = "Hathway, Arthur S.",
+ title = "A Primer Of Quaternions",
+ year = "1896"
+}
\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
+\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[Minc 79]{Min79} Henryk Minc
``Evaluation of Permanents''
Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 2732.
+\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{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[Hayes 74]{Hay74} Hayes J G.
+``Numerical Methods for Curve and Surface Fitting''
+Bull Inst Math Appl. 10 144152. (1974)
\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[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{chunk}{ignore}
\bibitem[Mitchell 80]{MG80} Mitchell A R.; Griffiths D F.
``The Finite Difference Method in Partial Differential Equations''
Wiley. (1980)
+\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
\end{chunk}
\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
+\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{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."
}
+\begin{chunk}{ignore}
+\bibitem[Higham 02]{Hig02} Higham, Nicholas J.
+``Accuracy and stability of numerical algorithms''
+SIAM Philadelphia, PA ISBN 0898715210 (2002)
\end{chunk}
\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[Hock 81]{HS81} Hock W.; Schittkowski K.
+``Test Examples for Nonlinear Programming Codes''
+Lecture Notes in Economics and Mathematical Systems. 187 SpringerVerlag. 1981
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Murray 72]{Mur72} Murray W, (ed)
``Numerical Methods for Unconstrained Optimization''
Academic Press. (1972)
+\bibitem[Householder 70]{Hou70} Householder A S.
+``The Numerical Treatment of a Single Nonlinear Equation''
+McGrawHill. (1970)
\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{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[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[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}
\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{I} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Nijenhuis 78]{NW78} Nijenhuis and Wilf
``Combinatorical Algorithms''
Academic Press, New York 1978.
+\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.
\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)

\end{chunk}

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

\begin{chunk}{axiom.bib}
@misc{OCAM14,
 author = "unknown",
 title = "The OCAML website",
 url = "http://ocaml.org"
}
+\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{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[Iyanaga 77]{Iya77} Iyanaga, Shokichi; Iyanaga, Yukiyosi Kawada
+``Encyclopedic Dictionary of Mathematics''
+1977
\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

\end{chunk}
+\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[OpenM]{OpenM}.
``OpenMath Technical Overview''
\verbwww.openmath.org/overview/technical.html
+\bibitem[Jacobson 68]{Jac68} Jacobson, N.
+``Structure and Representations of Jordan Algebras''
+AMS, Colloquium Publications Volume 39
\end{chunk}
\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[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}{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[Jaswon 77]{JS77} Jaswon, M A.; Symm G T.
+``Integral Equation Methods in Potential Theory and Elastostatics''
+Academic Press. (1977)
\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)
+\bibitem[Jeffrey 04]{Je04} Jeffrey, Alan
+``Handbook of Mathematical Formulas and Integrals''
+Third Edition, Elsevier Academic Press ISBN 0123822564
\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)
+\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}
+\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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[Kalkbrener 91]{Kal91} Kalkbrener, M.
+``Three contributions to elimination theory''
+Ph. D. Thesis, University of Linz, Austria, 1991
\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[Kalkbrener 98]{Kal98} Kalkbrener, M.
+``Algorithmic properties of polynomial rings''
+Journal of Symbolic Computation 1998
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Parlett 80]{Par80} Parlett B N.
``The Symmetric Eigenvalue Problem''
PrenticeHall. 1980
+\bibitem[Kantor 89]{Kan89} Kantor,I.L.; Solodovnikov, A.S.
+``Hypercomplex Numbers''
+Springer Verlag Heidelberg, 1989, ISBN 0387969802
\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[Kaufmann 00]{KMJ00} Kaufmann, Matt; Manolios, Panagiotis;
+Moore J Strother
+``ComputerAided Reasoning: An Approach''
+Springer, July 31. 2000 ISBN 0792377443
\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[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[Paul 81]{Paul81} Paul, Richard
``Robot Manipulators''
MIT Press 1981
+\bibitem[Knuth 84]{Knu84} Knuth, Donald
+{\it The \TeX{}book}.
+Reading, Massachusetts, AddisonWesley Publishing Company, Inc.,
+1984. ISBN 0201134489
\end{chunk}
\begin{chunk}{axiom.bib}
@book{Pear56,
 author = "Pearcey, T.",
 title = "Table of the Fresnel Integral",
 publisher = "Cambridge University Press",
 year = "1956"
}
+@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{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)
+\bibitem[Knu98]{Knu98} Donald Knuth
+``The Art of Computer Programming'' Vol. 3
+(Sorting and Searching)
+AddisonWesley 1998
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Peters 67a]{Pet67a} Peters G.
``NPL Algorithms Library''
Document No. F2/03/A. (1967)
+\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{chunk}{ignore}
\bibitem[Peters 67b]{Pet67b} Peters G.
``NPL Algorithms Library''
Document No.F1/04/A (1967)
+\bibitem[Kolchin 73]{Kol73} Kolchin, E.R.
+``Differential Algebra and Algebraic Groups''
+(Academic Press, 1973).
\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[Koutschan 10]{Kou10} Koutschan, Christoph
+``Axiom / FriCAS''
+\verbwww.risc.jku.at/education/courses/ws2010/cas/axiom.pdf
\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[Kozen 86]{KL86} Kozen, Dexter; Landau, Susan
+``Polynomial Decomposition Algorithms''
+Journal of Symbolic Computation (1989) 7, 445456
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Pierce 82]{Pie82} R.S. Pierce
``Associative Algebras''
Graduate Texts in Mathematics 88
SpringerVerlag, Heidelberg, 1982, ISBN 0387906932
+\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\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[Piessens 73]{Pie73} Piessens R.
``An Algorithm for Automatic Integration''
Angewandte Informatik. 15 399401. (1973)
+\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[Piessens 74]{PMB74} Piessens R.;; Mertens I.; Branders M.
``Integration of Functions having Endpoint Singularities''
Angewandte Informatik. 16 6568. (1974)
+\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{chunk}{ignore}
\bibitem[Piessens 75]{PB75} Piessens R.; Branders M.
``Algorithm 002. Computation of Oscillating Integrals''
J. Comput. Appl. Math. 1 153164. (1975)
+\bibitem[Lawson 74]{LH74} Lawson C L.; Hanson R J.
+``Solving Leastsquares Problems''
+PrenticeHall. (1974)
\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)
+\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"
+}
\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[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{chunk}{ignore}
\bibitem[Polya 37]{Pol37} Polya, G.
``Kombinatorische Anzahlbestimmungen fur Gruppen,
Graphen und chemische Verbindungen''
Acta Math. 68 (1937) 145254.
+\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[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[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{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}{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
+\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"
+}
\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[Lenstra 87]{LS87} Lenstra, H. W.; Schoof, R. J.
+``Primitivive Normal Bases for Finite Fields''
+Math. Comp. 48, 1987, pp. 217231
\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"
+@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{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
+\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{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)
+\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{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[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{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[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}{axiom.bib}
@misc{Puff09,
 author = "Puffinware LLC",
 title = "Singular Value Decomposition (SVD) Tutorial",
 url = "http://www.puffinwarellc.com/p3a.htm"
+@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}
\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.

\end{chunk}

\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Rabinowitz 70]{Rab70} Rabinowitz P.
``Numerical Methods for Nonlinear Algebraic Equations''
Gordon and Breach. (1970)
+\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{chunk}{ignore}
\bibitem[Ralston 65]{Ral65} Ralston A.
``A First Course in Numerical Analysis''
McGrawHill. 8790. (1965)
+\begin{chunk}{axiom.bib}
+@book{Losc60,
+ author = {L\"osch, Friedrich},
+ title = "Tables of Higher Functions",
+ publisher = "McGrawHill Book Company",
+ year = "1960"
+}
\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[LTU10]{LTU10}.
+``Lambda the Ultimate''
+\verblambdatheultimate.org/node/3663#comment62440
\end{chunk}
\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}
+@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{chunk}{ignore}
\bibitem[Redfield 27]{Red27} Redfield, J.H.
``The Theory of GroupReduced Distributions''
American J. Math., 49 (1927) 433455.
+\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{chunk}{ignore}
\bibitem[Reinsch 67]{Rei67} Reinsch C H.
``Smoothing by Spline Functions''
Num. Math. 10 177183. (1967)
+\bibitem[Lyness 83]{Lyn83} Lyness J N.
+``When not to use an automatic quadrature routine''
+SIAM Review. 25 6387. (1983)
\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)

\end{chunk}
+\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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[Mac Lane 79]{MB79} Mac Lane, Saunders; Birkhoff, Garret
+``Algebra''
+AMS Chelsea Publishing ISBN 0821816462
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Reutenauer 93]{Re93} Reutenauer, Christophe
``Free Lie Algebras''
Oxford University Press, June 1993 ISBN 0198536798
+\bibitem[Malcolm 72]{Mal72} Malcolm M. A.
+``Algorithms to reveal properties of floatingpoint arithmetic''
+Comms. of the ACM, 15, 949951. (1972)
\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[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{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
 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."
+\bibitem[Marden 66]{Mar66} Marden M.
+``Geometry of Polynomials''
+Mathematical Surveys. 3 Am. Math. Soc., Providence, RI. (1966)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rich 10]{Ri10} Rich, Albert D.
``Rulebased Mathematics''
\verbwww.apmaths.uwo.ca/~arich
+\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{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[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{chunk}{ignore}
\bibitem[Richtmyer 67]{RM67} Richtmyer R D.; Morton K W.
``Difference Methods for Initialvalue Problems''
Interscience (2nd Edition). (1967)
+\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{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[Maza 98]{Maz98} Maza, M. Moreno
+``A new algorithm for computing triangular
+decomposition of algebraic varieties''
+ NAG Tech. Rep. 4/98.
\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[Mignotte 82]{Mig82} Mignotte, Maurice
+``Some Useful Bounds''
+Computing, Suppl. 4, 259263 (1982), SpringerVerlag
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Ritt 50]{Ritt50} Ritt, Joseph Fels
``Differential Algebra''
AMS Colloquium Publications Volume 33 ISBN 9780821846384
+\bibitem[McCarthy 83]{McC83} McCarthy G J.
+``Investigation into the Multigrid Code MGD1''
+Report AERER 10889. Harwell. (1983)
\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[Mie97]{Mie97} Mielenz, Klaus D.
+``Computation of Fresnel Integrals''
+J. Res. Natl. Inst. Stand. Technol. (NIST) V102 No3 MayJune 1997 pp363365
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Rubey 07]{Rub07} Rubey, Martin
``Formula Guessing with Axiom''
April 2007
+\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{chunk}{ignore}
\bibitem[Rutishauser 69]{Rut69} Rutishauser H.
``Computational aspects of F L Bauer's simultaneous iteration method''
Num. Math. 13 413. (1969)
+\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{chunk}{ignore}
\bibitem[Rutishauser 70]{Rut70} Rutishauser H.
``Simultaneous iteration method for symmetric matrices''
Num. Math. 16 205223. (1970)
+\bibitem[Minc 79]{Min79} Henryk Minc
+``Evaluation of Permanents''
+Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 2732.
\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[More 74]{MGH74} More J J.; Garbow B S.; Hillstrom K E.
+``User Guide for Minpack1''
+ANL8074 Argonne National Laboratory. (1974)
\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[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[Schoenhage 82]{Sch82} Schoenhage, A.
``The fundamental theorem of algebra in terms of computational complexity''
preliminary report, Univ. Tuebingen, 1982
+\bibitem[Mitchell 80]{MG80} Mitchell A R.; Griffiths D F.
+``The Finite Difference Method in Partial Differential Equations''
+Wiley. (1980)
\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[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{chunk}{axiom.bib}
@book{Segg93,
 author = "{von Seggern}, David Henry",
 title = "CRC Standard Curves and Surfaces",
 publisher = "CRC Press",
 year = "1993",
 isbn = "0849301963"
+@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."
}
\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[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{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[Murray 72]{Mur72} Murray W, (ed)
+``Numerical Methods for Unconstrained Optimization''
+Academic Press. (1972)
\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
 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."
+\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{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[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}
+\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\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",
 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."
+\bibitem[Nijenhuis 78]{NW78} Nijenhuis and Wilf
+``Combinatorical Algorithms''
+Academic Press, New York 1978.
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Sit 92]{REFSit92} Sit, William
``An Algorithm for Parametric Linear Systems''
J. Sym. Comp., April 1992
+\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}
+
+\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@misc{OCAM14,
+ author = "unknown",
+ title = "The OCAML website",
+ url = "http://ocaml.org"
+}
\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[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[Smith 85]{Smi85} Smith G D.
``Numerical Solution of Partial Differential Equations: Finite Difference
Methods''
Oxford University Press (3rd Edition). (1985)
+\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{chunk}{ignore}
\bibitem[Sobol 74]{Sob74} Sobol I M.
``The Monte Carlo Method''
The University of Chicago Press. 1974
+\bibitem[OpenM]{OpenM}.
+``OpenMath Technical Overview''
+\verbwww.openmath.org/overview/technical.html
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Steele 90]{Ste90} Steele, Guy L.
``Common Lisp The Language''
Second Edition ISBN 1555580416 Digital Press (1990)
+\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}
@misc{Stic93,
 author = "Stichtenoth, H.",
 title = "Algebraic function fields and codes",
 publisher = "SpringerVerlag",
 year = "1993"
+@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{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
 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}
+\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
\bibitem[Stroud 66]{SS66} Stroud A H.; Secrest D.
``Gaussian Quadrature Formulas''
PrenticeHall. (1966)
+\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{chunk}{ignore}
\bibitem[Stroud 71]{Str71} Stroud A H.
``Approximate Calculation of Multiple Integrals''
PrenticeHall 1971
+\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{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[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[Swarztrauber 84]{SS84} Swarztrauber P N.
``Fast Poisson Solvers''
Studies in Numerical Analysis. (ed G H Golub)
Mathematical Association of America. (1984)

\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"
}
+\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{chunk}{ignore}
\bibitem[Taivalsaari 96]{Tai96} Taivalsaari, Antero
``On the Notion of Inheritance''
ACM Computing Surveys, Vol 28 No 3 Sept 1996 pp438479
+\bibitem[Parlett 80]{Par80} Parlett B N.
+``The Symmetric Eigenvalue Problem''
+PrenticeHall. 1980
\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[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{chunk}{ignore}
\bibitem[Temperton 83a]{Tem83a} Temperton C.
``Selfsorting Mixedradix Fast Fourier Transforms''
J. Comput. Phys. 52 123. (1983)
+\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[Temperton 83b]{Tem83b} Temperton C.
``Fast MixedRadix Real Fourier Transforms''
J. Comput. Phys. 52 340350. (1983)
+\bibitem[Paul 81]{Paul81} Paul, Richard
+``Robot Manipulators''
+MIT Press 1981
\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"
+@book{Pear56,
+ author = "Pearcey, T.",
+ title = "Table of the Fresnel Integral",
+ publisher = "Cambridge University Press",
+ year = "1956"
}
\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[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}
\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[Peters 67a]{Pet67a} Peters G.
+``NPL Algorithms Library''
+Document No. F2/03/A. (1967)
\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[Peters 67b]{Pet67b} Peters G.
+``NPL Algorithms Library''
+Document No.F1/04/A (1967)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Van Loan 92]{Van92} Van Loan, C.
``Computational Frameworks for the Fast Fourier Transform''
SIAM Philadelphia. (1992)
+\bibitem[Peters 70]{PW70} Peters G.; Wilkinson J H.
+``The Leastsquares Problem and Pseudoinverses''
+Comput. J. 13 309316. (1970)
\end{chunk}
\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{ignore}
\bibitem[Wait 85]{WM85} Wait R.; Mitchell A R.
``Finite Element Analysis and Application''
Wiley. (1985)
+\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{chunk}{ignore}
\bibitem[Wang 92]{Wang92} Wang, D.M.
``An implementation of the characteristic set method in Maple''
Proc. DISCO'92 Bath, England
+\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[Ward 75]{War75} Ward, R C.
``The Combination Shift QZ Algorithm''
SIAM J. Numer. Anal. 12 835853. 1975

\end{chunk}

\begin{chunk}{axiom.bib}
@misc{Watt03,
 author = "Watt, Stephen",
 title = "Aldor",
 url = "http://www.aldor.org",
 year = "2003"
}
+\bibitem[Piessens 73]{Pie73} Piessens R.
+``An Algorithm for Automatic Integration''
+Angewandte Informatik. 15 399401. (1973)
\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[Piessens 74]{PMB74} Piessens R.;; Mertens I.; Branders M.
+``Integration of Functions having Endpoint Singularities''
+Angewandte Informatik. 16 6568. (1974)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Weisstein]{Wein} Weisstein, Eric W.
``Hypergeometric Function''
MathWorld  A Wolfram Web Resource
\verbmathworld.wolfram.com/HypergeometricFunction.html
+\bibitem[Piessens 75]{PB75} Piessens R.; Branders M.
+``Algorithm 002. Computation of Oscillating Integrals''
+J. Comput. Appl. Math. 1 153164. (1975)
\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[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}{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[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[Wesseling 82a]{Wes82a} Wesseling, P.
``MGD1  A Robust and Efficient Multigrid Method''
Multigrid Methods. Lecture Notes in Mathematics. 960
SpringerVerlag. 614630. (1982)
+\bibitem[Polya 37]{Pol37} Polya, G.
+``Kombinatorische Anzahlbestimmungen fur Gruppen,
+Graphen und chemische Verbindungen''
+Acta Math. 68 (1937) 145254.
\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[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[Wicks 89]{Wic89} Wicks, Mark; Carlisle, David, Rahtz, Sebastian
``dvipdfm.def''
\verbweb.mit.edu/texsrc/source/latex/graphics/dvipdfm.def
+\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{chunk}{ignore}
\bibitem[Wiki 3]{Wiki3}.
``Givens Rotations''
\verben.wikipedia.org/wiki/Givens_rotation
+\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{chunk}{axiom.bib}
@misc{Wiki14a,
 author = "ProofWiki",
 title = "Euclidean Algorithm",
 url = "http://proofwiki.org/wiki/Euclidean_Algorithm"
+@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}{axiom.bib}
@misc{Wiki14b,
 author = "ProofWiki",
 title = "Division Theorem",
 url = "http://proofwiki.org/wiki/Division_Theorem"
}
+\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[Williamson 85]{Wil85} Williamson, S.G.
``Combinatorics for Computer Science''
Computer Science Press, 1985.
+\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{chunk}{ignore}
\bibitem[Wilkinson 71]{WR71} Wilkinson J H.; Reinsch C.
``Handbook for Automatic Computation II, Linear Algebra''
SpringerVerlag. 1971
+\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[Wilkinson 63]{Wil63} Wilkinson J H.
``Rounding Errors in Algebraic Processes''
 Chapter 2. HMSO. (1963)
+\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}
+@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[Wilkinson 65]{Wil65} Wilkinson J H.
``The Algebraic Eigenvalue Problem''
 Oxford University Press. (1965)
+\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[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[Rabinowitz 70]{Rab70} Rabinowitz P.
+``Numerical Methods for Nonlinear Algebraic Equations''
+Gordon and Breach. (1970)
\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[Ralston 65]{Ral65} Ralston A.
+``A First Course in Numerical Analysis''
+McGrawHill. 8790. (1965)
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wisbauer 91]{Wis91} Wisbauer, R.
``Bimodule Structure of Algebra''
Lecture Notes Univ. Duesseldorf 1991
+\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{chunk}{ignore}
\bibitem[WoerzBusekros 80]{Woe80} WoerzBusekros, A.
``Algebra in Genetics''
Lectures Notes in Biomathematics 36, SpringerVerlag, Heidelberg, 1980
+\bibitem[Ramsey 03]{Ra03} Ramsey, Norman
+``NowebA Simple, Extensible Tool for Literate Programming''
+\verbwww.eecs.harvard.edu/~nr/noweb
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wolberg 67]{Wol67} Wolberg J R.
``Prediction Analysis''
Van Nostrand. (1967)
+\bibitem[Redfield 27]{Red27} Redfield, J.H.
+``The Theory of GroupReduced Distributions''
+American J. Math., 49 (1927) 433455.
\end{chunk}
\begin{chunk}{ignore}
\bibitem[Wolfram 09]{Wo09} Wolfram Research
\verbmathworld.wolfram.com/Quaternion.html
+\bibitem[Reinsch 67]{Rei67} Reinsch C H.
+``Smoothing by Spline Functions''
+Num. Math. 10 177183. (1967)
\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[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[Wynn 56]{Wynn56} Wynn P.
``On a Device for Computing the $e_m(S_n )$ Transformation''
Math. Tables Aids Comput. 10 9196. (1956)
+\bibitem[Renka 84]{RC84} Renka R L.; Cline A K.
+``A Trianglebased C Interpolation Method''
+Rocky Mountain J. Math. 14 223237. (1984)
\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
 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."
+\bibitem[Reutenauer 93]{Re93} Reutenauer, Christophe
+``Free Lie Algebras''
+Oxford University Press, June 1993 ISBN 0198536798
\end{chunk}
\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[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[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[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 = "
 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."
+ 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}
\subsection{To Be Classified} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{chunk}{axiom.bib}
@PhdThesis{Kalt82,
 author = "Kaltofen, E.",
 title = "On the complexity of factoring polynomials with integer
 coefficients",
 school = "RPI",
 address = "Troy, N. Y.",
 year = "1982",
 month = "December",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_thesis.pdf",
 paper = "Kalt82.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Rich 10]{Ri10} Rich, Albert D.
+``Rulebased Mathematics''
+\verbwww.apmaths.uwo.ca/~arich
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt82a,
 author = "Kaltofen, E.",
 title = "A polynomialtime reduction from bivariate to univariate
 integral polynomial factorization",
 booktitle = "Proc. 23rd Annual Symp. Foundations of Comp. Sci.",
 year = "1982",
 pages = "5764",
 organization = "IEEE",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_focs.pdf",
 paper = "Kalt82a.pdf"
}
+\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
\end{chunk}
\begin{chunk}{axiom.bib}
@InCollection{Kalt82b,
 author = "Kaltofen, E.",
 title = "Polynomial factorization",
 editor = "B. Buchberger and G. Collins and R. Loos",
 booktitle = "Computer Algebra",
 edition = "2",
 pages = "95113",
 publisher = "SpringerVerlag",
 year = "1982",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_survey.ps.gz",
 keywords = "survey",
 paper = "Kalt82b.ps"
}
+\begin{chunk}{ignore}
+\bibitem[Richtmyer 67]{RM67} Richtmyer R D.; Morton K W.
+``Difference Methods for Initialvalue Problems''
+Interscience (2nd Edition). (1967)
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt83,
 author = "Kaltofen, E.",
 title = "On the complexity of finding short vectors in integer lattices",
 booktitle = "Proc. EUROCAL '83",
 series = "Lect. Notes Comput. Sci.",
 year = "1983",
 volume = "162",
 pages = "236244",
 publisher = "SpringerVerlag",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/83/Ka83_eurocal.pdf",
 paper = "Kalt83.pdf"
}
+\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
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt84,
 author = "Kaltofen, E.",
 title = "A Note on the {Risch} Differential Equation",
 booktitle = "Proc. EUROSAM '84",
 pages = "359366",
 crossref = "EUROSAM84",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_risch.ps.gz",
 paper = "Kalt84.ps"
}
+\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.
\end{chunk}
\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",
 booktitle = "Proc. EUROSAM '84",
 pages = "310320",
 crossref = "EUROSAM84",
 url =
 "http://www.math.ncsu.edu/~kaltofen/bibliography/84/KaYui84_eurosam.ps.gz",
 paper = "Kalt84a.ps"
}
+\begin{chunk}{ignore}
+\bibitem[Ritt 50]{Ritt50} Ritt, Joseph Fels
+``Differential Algebra''
+AMS Colloquium Publications Volume 33 ISBN 9780821846384
\end{chunk}
\begin{chunk}{axiom.bib}
@TechReport{Kalt84b,
 author = "Kaltofen, E.",
 title = "The Algebraic Theory of Integration",
 institution = "RPI",
 address = "Dept. Comput. Sci., Troy, New York",
 year = "1984",
 url =
 "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_integration.pdf",
 paper = "Kalt84b.pdf"
}

\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
\begin{chunk}{axiom.bib}
@InProceedings{Kalt85,
 author = "Kaltofen, E.",
 title = "Effective {Hilbert} Irreducibility",
 booktitle = "Proc. EUROSAM '84",
 pages = "275284",
 crossref = "EUROSAM84",
 url =
 "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_infcontr.ps.gz",
 paper = "Kalt85.ps"
}

\end{chunk}
\begin{chunk}{axiom.bib}
@Article{Kalt85a,
 author = "Kaltofen, E.",
 title = "Fast parallel absolute irreducibility testing",
 journal = "Journal of Symbolic Computation",
 year = "1985",
 volume = "1",
 number = "1",
 pages = "5767",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_jsc.pdf",
 paper = "Kalt85a.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Rubey 07]{Rub07} Rubey, Martin
+``Formula Guessing with Axiom''
+April 2007
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt85b,
 author = "Kaltofen, E.",
 title = "Computing with polynomials given by straightline programs {II};
 sparse factorization",
 booktitle = "Proc. 26th Annual Symp. Foundations of Comp. Sci.",
 year = "1985",
 pages = "451458",
 organization = "IEEE",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_focs.ps.gz",
 paper = "Kalt85b.ps"
}
+\begin{chunk}{ignore}
+\bibitem[Rutishauser 69]{Rut69} Rutishauser H.
+``Computational aspects of F L Bauer's simultaneous iteration method''
+Num. Math. 13 413. (1969)
\end{chunk}
\begin{chunk}{axiom.bib}
@TechReport{Kalt85c,
 author = "E. Kaltofen",
 title = "Sparse Hensel lifting",
 institution = "RPI",
 address = "Dept. Comput. Sci., Troy, N. Y.",
 year = "1985",
 number = "8512",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_techrep.pdf",
 paper = "Kalt85c.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Rutishauser 70]{Rut70} Rutishauser H.
+``Simultaneous iteration method for symmetric matrices''
+Num. Math. 16 205223. (1970)
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt85d,
 author = "Kaltofen, E.",
 title = "Sparse Hensel lifting",
 booktitle = "EUROCAL 85 European Conf. Comput. Algebra Proc. Vol. 2",
 crossref = "EUROCAL85",
 pages = "417",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_eurocal.pdf",
 paper = "Kalt85d.pdf"
}
+\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\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}
@Article{Kalt85e,
 author = "Kaltofen, E.",
 title = "Polynomialtime reductions from multivariate to bi and univariate
 integral polynomial factorization",
 journal = "{SIAM} J. Comput.",
 year = "1985",
 volume = "14",
 number = "2",
 pages = "469489",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_sicomp.pdf",
 paper = "Kalt85e.pdf"
}
+\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{Gath85,
 author = "Gathen, Joachim von zur; Kaltofen, E.",
 title = "Factoring sparse multivariate polynomials",
 journal = "J. Comput. System Sci.",
 year = "1985",
 volume = "31",
 pages = "265287",
 url =
 "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz",
 paper = "Gath85.ps"
}
+\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{chunk}{axiom.bib}
@InProceedings{Kalt86,
 author = "Kaltofen, E.",
 title = "Uniform closure properties of pcomputable functions",
 booktitle = "Proc. 18th Annual ACM Symp. Theory Comput.",
 year = "1986",
 pages = "330337",
 organization = "ACM",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/86/Ka86_stoc.pdf",
 paper = "Kalt86.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{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",
 journal = "SIAM J. Alg. Discrete Math.",
 year = "1987",
 volume = "8",
 pages = "683690",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/KKS87.pdf",
 paper = "Kalt87.pdf"
+@book{Segg93,
+ author = "{von Seggern}, David Henry",
+ title = "CRC Standard Curves and Surfaces",
+ publisher = "CRC Press",
+ year = "1993",
+ isbn = "0849301963"
}
\end{chunk}
\begin{chunk}{axiom.bib}
@InCollection{Kalt87a,
 author = "Kaltofen, E.",
 editor = "J. F. Traub",
 title = "Computer algebra algorithms",
 booktitle = "Annual Review in Computer Science",
 pages = "91118",
 publisher = "Annual Reviews Inc.",
 year = "1987",
 volume = "2",
 address = "Palo Alto, California",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_annrev.pdf",
 paper = "Kalt87a.pdf"
}
+\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
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt87b,
 author = "Kaltofen, E.",
 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",
 organization = "ACM",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_stoc.pdf",
 paper = "Kalt87b.pdf"
}
+\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
\end{chunk}
\begin{chunk}{axiom.bib}
@Article{Kalt87c,
 author = "Kaltofen, E.",
 title = "Deterministic irreducibility testing of polynomials over
 large finite fields",
 journal = "Journal of Symbolic Computation",
 year = "1987",
 volume = "4",
 pages = "7782",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_jsc.ps.gz",
 paper = "Kalt87c.ps"
}
+\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}{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",
 booktitle = "Proc. 29th Annual Symp. Foundations of Comp. Sci.",
 pages = "296305",
 year = "1988",
 organization = "IEEE",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/focs88.ps.gz",
 paper = "Kalt88.ps"
}
+\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
\end{chunk}
\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",
 journal = "SIAM J. Comput.",
 year = "1988",
 volume = "17",
 number = "4",
 pages = "687695",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/MRK88.pdf",
 paper = "Mill88.pdf"
}
+\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",
+ 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{chunk}{axiom.bib}
@InProceedings{Kalt88a,
 author = "Kaltofen, E. and Yagati, Lakshman",
 title = "Improved sparse multivariate polynomial interpolation algorithms",
 booktitle = "Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc.",
 crossref = "ISSAC88",
 pages = "467474",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/KaLa88.pdf",
 paper = "Kalt88a.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Sit 92]{REFSit92} Sit, William
+``An Algorithm for Parametric Linear Systems''
+J. Sym. Comp., April 1992
\end{chunk}
\begin{chunk}{axiom.bib}
@Article{Kalt88b,
 author = "Kaltofen, E.",
 title = "Greatest common divisors of polynomials given by
 straightline programs",
 journal = "J. ACM",
 year = "1988",
 volume = "35",
 number = "1",
 pages = "231264",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/Ka88_jacm.pdf",
 paper = "Kalt88b.pdf"
}
+\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}{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",
 journal = "ACM Trans. Math. Software",
 year = "1988",
 volume = "14",
 number = "3",
 pages = "218240",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/FIKY88.pdf",
 paper = "Free88.pdf"
}
+\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)
\end{chunk}
\begin{chunk}{axiom.bib}
@Article{Greg88,
 author = "Gregory, B.; Kaltofen, E.",
 title = "Analysis of the binary complexity of asymptotically fast
 algorithms for linear system solving",
 journal = "SIGSAM Bulletin",
 year = "1988",
 month = "April",
 volume = "22",
 number = "2",
 pages = "4149",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/GrKa88.pdf",
 paper = "Grey88.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Sobol 74]{Sob74} Sobol I M.
+``The Monte Carlo Method''
+The University of Chicago Press. 1974
\end{chunk}
\begin{chunk}{axiom.bib}
@InCollection{Kalt89,
 author = "Kaltofen, E.",
 editor = "S. Micali",
 title = "Factorization of polynomials given by straightline programs",
 booktitle = "Randomness and Computation",
 pages = "375412",
 publisher = "JAI Press Inc.",
 year = "1989",
 volume = "5",
 series = "Advances in Computing Research",
 address = "Greenwhich, Connecticut",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_slpfac.pdf",
 paper = "Kalt89.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Steele 90]{Ste90} Steele, Guy L.
+``Common Lisp The Language''
+Second Edition ISBN 1555580416 Digital Press (1990)
\end{chunk}
\begin{chunk}{axiom.bib}
@Article{Kalt89a,
 author = "Kaltofen, E.; Rolletschek, H.",
 title = "Computing greatest common divisors and factorizations in
 quadratic number fields",
 journal = "Math. Comput.",
 year = "1989",
 volume = "53",
 number = "188",
 pages = "697720",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KaRo89.pdf",
 paper = "Kalt89a.pdf"
+@misc{Stic93,
+ author = "Stichtenoth, H.",
+ title = "Algebraic function fields and codes",
+ publisher = "SpringerVerlag",
+ year = "1993"
}
\end{chunk}
\begin{chunk}{axiom.bib}
@Unpublished{Kalt89b,
 author = "Kaltofen, E.",
 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",
 paper = "Kalt89b.ps"
}
+\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
+ 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}{axiom.bib}
@TechReport{Kalt89c,
 author = "Kaltofen, E.",
 title = "Parallel Algebraic Algorithm Design",
 institution = "RPI",
 address = "Dept. Comput. Sci., Troy, New York",
 year = "1989",
 month = "July",
 url =
 "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_parallel.ps.gz",
 paper = "Kalt89c.ps"
}
+\begin{chunk}{ignore}
+\bibitem[Stroud 66]{SS66} Stroud A H.; Secrest D.
+``Gaussian Quadrature Formulas''
+PrenticeHall. (1966)
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Cann89,
 author = "Canny, J. and Kaltofen, E. and Yagati, Lakshman",
 title = "Solving systems of nonlinear polynomial equations faster",
 booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.",
 crossref = "ISSAC89",
 pages = "121128",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/CKL89.pdf",
 paper = "Cann89.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Stroud 71]{Str71} Stroud A H.
+``Approximate Calculation of Multiple Integrals''
+PrenticeHall 1971
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt89d,
 author = "Kaltofen, E. and Valente, T. and Yui, N.",
 title = "An improved {Las Vegas} primality test",
 booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.",
 crossref = "ISSAC89",
 pages = "2633",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KVY89.pdf",
 paper = "Kalt89d.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{chunk}{axiom.bib}
@InProceedings{Kalt90,
 author = "Kaltofen, E. and Lakshman, Y.N. and Wiley, J.M.",
 editor = "S. Watanabe and M. Nagata",
 title = "Modular rational sparse multivariate polynomial interpolation",
 booktitle = "Proc. 1990 Internat. Symp. Symbolic Algebraic Comput.",
 pages = "135139",
 publisher = "ACM Press",
 year = "1990",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KLW90.pdf",
 paper = "Kalt90.pdf"
}
+\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)
\end{chunk}
+\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{axiom.bib}
@Article{Kalt90a,
 author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.",
 title = "Parallel algorithms for matrix normal forms",
 journal = "Linear Algebra and Applications",
 year = "1990",
 volume = "136",
 pages = "189208",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KKS90.pdf",
 paper = "Kalt90a.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{chunk}{axiom.bib}
@Article{Kalt90b,
 author = "Kaltofen, E.",
 title = "Computing the irreducible real factors and components of an
 algebraic curve",
 journal = "Applic. Algebra Engin. Commun. Comput.",
 year = "1990",
 volume = "1",
 number = "2",
 pages = "135148",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_aaecc.pdf",
 paper = "Kalt90b.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Taivalsaari 96]{Tai96} Taivalsaari, Antero
+``On the Notion of Inheritance''
+ACM Computing Surveys, Vol 28 No 3 Sept 1996 pp438479
\end{chunk}
\begin{chunk}{axiom.bib}
@InCollection{Kalt90c,
 author = "Kaltofen, E.",
 editor = "D. V. Chudnovsky and R. D. Jenks",
 title = "Polynomial Factorization 19821986",
 booktitle = "Computers in Mathematics",
 pages = "285309",
 publisher = "Marcel Dekker, Inc.",
 year = "1990",
 volume = "125",
 series = "Lecture Notes in Pure and Applied Mathematics",
 address = "New York, N. Y.",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_survey.ps.gz",
 keywords = "survey",
 paper = "Kalt90c.ps"
}
+\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)
\end{chunk}
\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",
 journal = "J. Symbolic Comput.",
 year = "1990",
 volume = "9",
 number = "3",
 pages = "301320",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KaTr90.pdf",
 paper = "Kalt90d.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Temperton 83a]{Tem83a} Temperton C.
+``Selfsorting Mixedradix Fast Fourier Transforms''
+J. Comput. Phys. 52 123. (1983)
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt91,
 author = "Kaltofen, E. and Saunders, B.D.",
 editor = "H. F. Mattson and T. Mora and T. R. N. Rao",
 title = "On {Wiedemann's} method of solving sparse linear systems",
 booktitle = "Proc. AAECC9",
 series = "Lect. Notes Comput. Sci.",
 volume = "539",
 pages = "2938",
 publisher = "SpringerVerlag",
 year = "1991",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSa91.pdf",
 paper = "Kalt91.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Temperton 83b]{Tem83b} Temperton C.
+``Fast MixedRadix Real Fourier Transforms''
+J. Comput. Phys. 52 340350. (1983)
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt91a,
 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",
 pages = "133145",
 publisher = "World Scientific Publ. Co.",
 year = "1991",
 address = "Singapore",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSi91.pdf",
 paper = "Kalt91a.pdf"
+@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}
\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",
 booktitle = "Number Theory New York Seminar 19891990",
 pages = "150202",
 publisher = "SpringerVerlag",
 year = "1991",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaYui91.pdf",
 paper = "Kalt91b.pdf"
}
+\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
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Diaz91,
 author = "Diaz, A.; Kaltofen,E.; Schmitz, K.; Valente, T.",
 title = "DSC A System for Distributed Symbolic Computation",
 booktitle = "Proc. 1991 Internat. Symp. Symbolic Algebraic Comput.",
 crossref = "ISSAC91",
 pages = "323332",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/DKSV91.pdf",
 paper = "Diaz91.pdf"
}
+\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)
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt91c,
 author = "Kaltofen, E. and Pan, V.",
 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",
 year = "1991",
 address = "New York, N.Y.",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaPa91.pdf",
 paper = "Kalt91c.pdf"
}
+\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
\end{chunk}
\begin{chunk}{axiom.bib}
@Article{Cant91,
 author = "Cantor, D.G. and Kaltofen, E.",
 title = "On fast multiplication of polynomials over arbitrary algebras",
 journal = "Acta Inform.",
 year = "1991",
 volume = "28",
 number = "7",
 pages = "693701",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/CaKa91.pdf",
 paper = "Cant91.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Van Loan 92]{Van92} Van Loan, C.
+``Computational Frameworks for the Fast Fourier Transform''
+SIAM Philadelphia. (1992)
\end{chunk}
\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",
 booktitle = "Proc. 33rd Annual Symp. Foundations of Comp. Sci.",
 year = "1992",
 pages = "714723",
 publisher = "IEEE Computer Society Press",
 address = "Los Alamitos, California",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/KaPa92.pdf",
 paper = "Kalt92.pdf"
}
+\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Wait 85]{WM85} Wait R.; Mitchell A R.
+``Finite Element Analysis and Application''
+Wiley. (1985)
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt92a,
 author = "Kaltofen, E.",
 title = "On computing determinants of matrices without divisions",
 booktitle = "Proc. 1992 Internat. Symp. Symbolic Algebraic Comput.",
 crossref = "ISSAC92",
 pages = "342349",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_issac.pdf",
 paper = "Kalt92a.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{chunk}{axiom.bib}
@InProceedings{Kalt92b,
 author = "Kaltofen, E.",
 title = "Polynomial factorization 19871991",
 booktitle = "Proc. LATIN '92",
 editor = "I. Simon",
 series = "Lect. Notes Comput. Sci.",
 volume = "583",
 pages = "294313",
 publisher = "SpringerVerlag",
 year = "1992",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_latin.pdf",
 keywords = "survey",
 paper = "Kalt92b.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Ward 75]{War75} Ward, R C.
+``The Combination Shift QZ Algorithm''
+SIAM J. Numer. Anal. 12 835853. 1975
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt93,
 author = "Kaltofen, E.",
 title = "Computational Differentiation and Algebraic Complexity Theory",
 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",
 series = "Tech. Rep.",
 volume = "ANL/MCSTM183",
 month = "December",
 year = "1993",
 pages = "2830",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_diff.pdf",
 paper = "Kalt93.pdf"
+@misc{Watt03,
+ author = "Watt, Stephen",
+ title = "Aldor",
+ url = "http://www.aldor.org",
+ year = "2003"
}
\end{chunk}
\begin{chunk}{axiom.bib}
@InCollection{Kalt93a,
 author = "Kaltofen, E.",
 editor = "J. Reif",
 title = "Dynamic parallel evaluation of computation {DAG}s",
 booktitle = "Synthesis of Parallel Algorithms",
 pages = "723758",
 publisher = "Morgan Kaufmann Publ.",
 year = "1993",
 address = "San Mateo, California",
 url =
 "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_synthesis.ps.gz",
 paper = "Kalt93a.ps"
+@misc{Weil71,
+ author = "Weil, Andr\'{e}",
+ title = "Courbes alg\'{e}briques et vari\'{e}t\'{e}s Abeliennes",
+ year = "1971"
}
\end{chunk}
\begin{chunk}{axiom.bib}
@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",
 booktitle = "Proc. DISCO '93",
 series = "Lect. Notes Comput. Sci.",
 pages = "6680",
 year = "1993",
 volume = "722",
 publisher = "SpringerVerlag",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/DHKLV93.pdf",
 paper = "Diaz93.pdf"
}
+\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{Kalt93b,
 author = "Kaltofen, E.",
 title = "Direct proof of a theorem by Kalkbrener, Sweedler, and Taylor",
 journal = "SIGSAM Bulletin",
 year = "1993",
 volume = "27",
 number = "4",
 pages = "2",
 url =
 "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_sambull.ps.gz",
 paper = "Kalt93b.ps"
+@misc{Weit03,
+ author = "Weitz, E.",
+ title = "CLWHO Yet another Lisp markup language",
+ year = "2003",
+ url = "http://www.weitz.de/clwho/"
}
\end{chunk}
\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",
 booktitle = "Proc. First Internat. Symp. Parallel Symbolic Comput.",
 crossref = "PASCO94",
 pages = "225233",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaPa94.pdf",
 paper = "Kalt94.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{chunk}{axiom.bib}
@InProceedings{Chan94,
 author = "Chan, K.C. and Diaz, A. and Kaltofen, E.",
 editor = "R. J. Lopez",
 title = "A distributed approach to problem solving in Maple",
 booktitle = "Maple V: Mathematics and its Application",
 pages = "1321",
 publisher = {Birkh\"auser},
 year = "1994",
 series = "Proceedings of the Maple Summer Workshop and Symposium (MSWS'94)",
 address = "Boston",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/CDK94.ps.gz",
 paper = "Chan94.ps"
}
+\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)
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt94a,
 author = "Kaltofen, E. and Lobo, A.",
 title = "Factoring highdegree polynomials by the black box
 Berlekamp algorithm",
 booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
 crossref = "ISSAC94",
 pages = "9098",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaLo94.ps.gz",
 paper = "Kalt94a.ps"
}
+\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{chunk}{axiom.bib}
@InProceedings{Kalt94b,
 author = "Kaltofen, E.",
 title = "Asymptotically fast solution of {Toeplitz}like singular
 linear systems",
 booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
 pages = "297304",
 crossref = "ISSAC94",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/Ka94_issac.pdf",
 paper = "Kalt94b.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Wicks 89]{Wic89} Wicks, Mark; Carlisle, David, Rahtz, Sebastian
+``dvipdfm.def''
+\verbweb.mit.edu/texsrc/source/latex/graphics/dvipdfm.def
\end{chunk}
\begin{chunk}{axiom.bib}
@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",
 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",
 paper = "Sama95.ps"
}
+\begin{chunk}{ignore}
+\bibitem[Wiki 3]{Wiki3}.
+``Givens Rotations''
+\verben.wikipedia.org/wiki/Givens_rotation
\end{chunk}
\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",
 journal = "Math. Comput.",
 year = "1995",
 volume = "64",
 number = "210",
 pages = "777806",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_mathcomp.pdf",
 paper = "Kalt95.pdf"
+@misc{Wiki14a,
+ author = "ProofWiki",
+ title = "Euclidean Algorithm",
+ url = "http://proofwiki.org/wiki/Euclidean_Algorithm"
}
\end{chunk}
\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",
 booktitle = "Proc. 1995 Internat. Symp. Symbolic Algebraic Comput.",
 crossref = "ISSAC95",
 pages = "232239",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DiKa95.ps.gz",
 paper = "Diaz95.ps"
+@misc{Wiki14b,
+ author = "ProofWiki",
+ title = "Division Theorem",
+ url = "http://proofwiki.org/wiki/Division_Theorem"
}
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt95a,
 author = "Kaltofen, E. and Shoup, V.",
 title = "Subquadratictime factoring of polynomials over finite fields",
 booktitle = "Proc. 27th Annual ACM Symp. Theory Comput.",
 year = "1995",
 publisher = "ACM Press",
 address = "New York, N.Y.",
 pages = "398406",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/KaSh95.ps.gz",
 paper = "Kalt95a.ps"
}
+\begin{chunk}{ignore}
+\bibitem[Williamson 85]{Wil85} Williamson, S.G.
+``Combinatorics for Computer Science''
+Computer Science Press, 1985.
\end{chunk}
\begin{chunk}{axiom.bib}
@Article{Hitz95,
 author = "Kitz, M.A. and Kaltofen, E.",
 title = "Integer division in residue number systems",
 journal = "IEEE Trans. Computers",
 year = "1995",
 volume = "44",
 number = "8",
 pages = "983989",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/HiKa95.pdf",
 paper = "Hitz95.pdf"
}
+\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{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",
 journal = "Journal of Symbolic Computing",
 year = "1995",
 volume = "19",
 number = "13",
 pages = "269282",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DHKLV95.pdf",
 paper = "Diaz95a.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 63]{Wil63} Wilkinson J H.
+``Rounding Errors in Algebraic Processes''
+ Chapter 2. HMSO. (1963)
\end{chunk}
\begin{chunk}{axiom.bib}
@Article{Kalt95b,
 author = "Kaltofen, E.",
 title = "Effective {Noether} irreducibility forms and applications",
 journal = "J. Comput. System Sci.",
 year = "1995",
 volume = "50",
 number = "2",
 pages = "274295",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_jcss.pdf",
 paper = "Kalt95b.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 65]{Wil65} Wilkinson J H.
+``The Algebraic Eigenvalue Problem''
+ Oxford University Press. (1965)
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Erli96,
 author = "Erlingsson, U. and Kaltofen, E. and Musser, D.",
 title = "Generic {Gram}{Schmidt} Orthogonalization by Exact Division",
 booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.",
 crossref = "ISSAC96",
 pages = "275282",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/EKM96.pdf",
 paper = "Erli96.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{chunk}{axiom.bib}
@InProceedings{Kalt96,
 author = "Kaltofen, E. and Lobo, A.",
 title = "On rank properties of {Toeplitz} matrices over finite fields",
 booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.",
 crossref = "ISSAC96",
 pages = "241249",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_issac.pdf",
 paper = "Kalt96.pdf"
}
+\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
\end{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt96a,
 author = "Kaltofen, E. and Lobo, A.",
 title = "Distributed matrixfree solution of large sparse linear systems
 over finite fields",
 booktitle = "Proc. High Performance Computing '96",
 year = "1996",
 editor = "A. M. Tentner",
 pages = "244247",
 organization = "Society for Computer Simulation",
 publisher = "Simulation Councils, Inc.",
 address = "San Diego, CA",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_hpc.pdf",
 paper = "Kalt96a.pdf"
}
+\begin{chunk}{ignore}
+\bibitem[Wisbauer 91]{Wis91} Wisbauer, R.
+``Bimodule Structure of Algebra''
+Lecture Notes Univ. Duesseldorf 1991
\end{chunk}
\begin{chunk}{axiom.bib}
@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)",
 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",
 paper = "Kalt96b.ps"
}
+\begin{chunk}{ignore}
+\bibitem[WoerzBusekros 80]{Woe80} WoerzBusekros, A.
+``Algebra in Genetics''
+Lectures Notes in Biomathematics 36, SpringerVerlag, Heidelberg, 1980
\end{chunk}
\begin{chunk}{axiom.bib}
@Article{Kalt97,
 author = "E. Kaltofen",
 title = "Teaching Computational Abstract Algebra",
 journal = "Journal of Symbolic Computation",
 volume = "23",
 number = "56",
 pages = "503515",
 year = "1997",
 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",
+\begin{chunk}{ignore}
+\bibitem[Wolberg 67]{Wol67} Wolberg J R.
+``Prediction Analysis''
+Van Nostrand. (1967)
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Wolfram 09]{Wo09} Wolfram Research
+\verbmathworld.wolfram.com/Quaternion.html
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Wu 87]{WU87} Wu, W.T.
+``A Zero Structure Theorem for polynomial equations solving''
+MM Research Preprints, 1987
+
+\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}
+
+\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
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."
}
+ 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{chunk}
\begin{chunk}{axiom.bib}
@InProceedings{Kalt97a,
 author = "Kaltofen, E. and Shoup, V.",
 title = "Fast polynomial factorization over high algebraic extensions of
 finite fields",
 booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.",
 crossref = "ISSAC97",
 pages = "184188",
 url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/KaSh97.pdf",
 paper = "Kalt97a.pdf"
+@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}
+
\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Bibliography}
diff git a/changelog b/changelog
index 14f3319..0dac261 100644
 a/changelog
+++ b/changelog
@@ 1,3 +1,5 @@
+20140920 tpd src/axiomwebsite/patches.html 20140920.02.tpd.patch
+20140920 tpd books/bookvolbib add abstracts, rearrange, add new sections
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
diff git a/patch b/patch
index a644f10..73fef81 100644
 a/patch
+++ b/patch
@@ 1,3 +1,3 @@
books/bookvolbib add abstracts, rearrange, add new entries
+books/bookvolbib add abstracts, rearrange, add new sections
Expand and cleanup bibliography
diff git a/src/axiomwebsite/patches.html b/src/axiomwebsite/patches.html
index 5e0a8d3..6c4d579 100644
 a/src/axiomwebsite/patches.html
+++ b/src/axiomwebsite/patches.html
@@ 4648,6 +4648,8 @@ books/bookvolbib add references
books/axiom.bst use axiom specific bib style
20140920.01.tpd.patch
books/bookvolbib add abstracts, rearrange, add new entries
+20140920.02.tpd.patch
+books/bookvolbib add abstracts, rearrange, add new sections