# daly/axiom

Commits on Jun 29, 2016
1. Goal: Axiom Literate Programming

\index{Colin, Antoine}
\begin{chunk}{axiom.bib}
@article{Coli97,
author = "Colin, Antoine",
title = "Solving a system of algebraic equations with symmetries",
journal = "J. Pure Appl. Algebra",
volume = "117-118",
pages = "195-215",
year = "1997",
keywords = "axiomref",
abstract =
"Let $(F)$ be a system of $p$ polynomial equations
$F_i({\bf X}) \in k[{\bf X}]$, where $k$ is a commutative field and
${\bf X} := (X_1,\cdots,X_n)$ are indeterminates. Let $G$ be a subgroup
of $GL_n(k)$. A polynomial $P \in k[{\bf X}]$ (resp. rational function
$P \in k({\bf X})$ ) is an invariant of $G$ if and only if for all
$A \in G$ we have $A\cdot P = P$. We denote $k[{\bf X}]^G$ by (resp.
$k({\bf X})^G$) the algebra of polynomial (resp. rational function)
invariants of $G$. If $L$ is another subgroup of $GL_n(k)$ such that
$G \subset L$, $P$ is called a primary invariant of $G$ relative to $L$ if
and only if $Stab_L(P) = G$ (where $Stab_L(P)$ is the stabilizer of
$P$ in $L$).

The paper describes the algebra of the invariants of a finite group
and how to express these invariants in terms of a small number of
them, from both the Cohen-Macaulay algebra and the field theory points
of view. A method is proposed to solve $(F)$ by expressing it in terms of
primary invariants $\Pi_1,\cdots,\Pi_n$
(e.g. the elementary symmetric polynomials) and one
primitive'' secondary invariant.

The main thrust of the paper is contained in the following theorem.
Let $(F)$ be a set of invariants of $G$. Let $L$ be a subgroup of
$GL_n(k)$ such that $G \subset L$ and $k({\bf X})^L$ is a purely
transcendental extension of $k_i$, let $\Pi_1,\cdots,\Pi_n$ be
polynomials such that $k({\bf X})^L = k(\Pi_1,\cdots,\Pi_n)$,
and let $\Theta \in k[{\bf X}]^G$ be a primitive polynomial invariant
of $G$ relative to $L$.
When possible, it is convenient to choose $\Theta$ to be one of the
polynomials in $(F)$. – An algorithm is given that allows each polynomial
$F_i$ to be expressed as $F_i({\bf X}) = H_i(\Pi_1,\cdots,\Pi_n,\Theta)$,
an algebraic fraction in $\Pi_1,\cdots,\Pi_n$ and a polynomial in
$\Theta$. Now let $L$ be the minimal polynomial of $\Theta$ over
$k[{\bf X}]^L$; we have
$L({\bf X},T)=\prod_{\Theta^{'} \in L\cdot \Theta}(T-\Theta^{'}) \in k[{\bf X}]^L[T]$
(where $L$ is called a generic Lagrange resolvent).
As $k(\Pi_1,\cdots,\Pi_n)=k({\bf X})^L$, we can write
$L({\bf X},T)=H_0(\Pi_1,\cdots,\Pi_n,T)$ where $H_0$ is some
rational function. The question
$H_0(\Pi_1,\cdots,\Pi_n,\Theta)=0$ is always satisfied because
$\Theta$ is a root of $L$. Then, we solve the system of ($p=1$)
algebraic equations $H_i(\Pi_1,\cdots,\Pi_n,\Theta)=0$,
$0 \le i \le p$ for $\Pi_1,\cdots,\Pi_n,\Theta$ as indeterminates.

Theorem 1: Let $D \in k[\Pi_1,\cdots,\Pi_n]$ be the LCM of the
denominators of all the fractions $H_i$,$0 \le i \le p$ and let
$H_i^{'}=DH_i$. For every solution
$x:=(x_1,\cdots,x_n)$ of the system $(F)$:$F_i({\bf X})=0$,
$1 \le i \le p$, there exists a solution ($\pi_1,\cdots,\pi_n,\Theta$)
of the system
$(H^{'}):H_i^{'}(\Pi_1,\cdots,\Pi_n,\Theta)=0$, $0 \le i \le p$ such
that $x$ is a solution of the system
$(P_\pi):\Pi_i({\bf X})=\pi_i$, $1 \le i \le n$ , and of the equation
$\Theta({\bf X})=0$. Conversely, for any solution
$($i_1,\cdots,\pi_n,\theta) of the system (H^{'}) such that D(\pi_1,\cdots,\pi_n) \ne 0, if x is a solution of the system (P_\pi) relative to (\pi_1,\cdots,\pi_n), then there exists some A \in L such that \Theta(A\cdot x)=\theta, and then for all B \in G, BA\cdot x, is a solution of the system (F). A slighly more general version of this theorem is also given. The paper then presents an algorithm that applies the theory and has been implemented in AXIOM. It is followed by several examples." } \end{chunk} \index{DiBlasio, Paolo} \index{Temperini, Marco} \begin{chunk}{axiom.bib} @article{DiBl95, author = "DiBlasio, Paolo and Temperini, Marco", title = "Subtyping Inheritance and Its Application in Languages for Symbolic Computation Systems", journal = "J. Symbolic Computation", volume = "19", pages = "39-63", year = "1995", paper = "DiBl95.pdf", keywords = "axiomref", abstract = "Application of object-oriented programming techniques to design and implementation of symbolic computation is investigated. We show the significance of certain correctness problems, occurring in programming environments based on specialization inheritance, due to use of method redefinition and polymorphism. We propose a solution to these problems, by defining a mechanism of subtyping inheritance and the prototype of an object-oriented programming language for a symbolic computation system. We devise the subtyping inheritance {\sl ESI (Enhanced String Inheritance)} by lifting to programming language constructs a given model of subtyping, which is established by a monotonic (covariant) subtyping rule. Type safeness of language instructions is proved. The adoption of {\sl ESI} allows to model method and class specialization in a natural way. The {\sl ESI} mechanism verifies the type correctness of language statements by means of type checking rules and preserves their correctness at run-time by a suitable method lookup algorithm." } \end{chunk} \index{DiBlasio, Paolo} \index{Temperini, Marco} \begin{chunk}{axiom.bib} @InProceedings{DiBl97, author = "DiBlasio, Paolo and Temperini, Marco", title = "On subtyping in languages for symbolic computation systems", booktitle = "Advances in the design of symbolic computation systems", series = "Monographs in Symbolic Computation", year = "1997", publisher = "Springer", pages = "164-178", keywords = "axiomref", abstract = "We want to define a strongly typed OOP language suitable as the software development tool of a symbolic computation system, which provides class structure to manage ADTs and supports multiple inheritance to model specialization hierarchies. In this paper, we provide the theoretical background for such a task." } \end{chunk} \index{Fakler, Winfried} \begin{chunk}{axiom.bib} @article{Fakl97, author = "Fakler, Winfried", title = "On second order homogeneous linear differential equations with Liouvillian solutions", journal = "Theor. Comput. Sci.", volume = "187", number = "1-2", pages = "27-48", year = "1997", paper = "Fakl97.pdf", keywords = "axiomref", abstract = "We determine all minimal polynomials for second order homogeneous linear differential equations with algebraic solutions decomposed into invariants and we show how easily one can recover the known conditions on differential Galois groups [J. Kovacic, J. Symb. Comput. 2, 3-43 (1986; Zbl 0603.68035), M. F. Singer and F. Ulmer, J. Symb. Comput. 16, 9-36, 37-73 (1993; Zbl 0802.12004, Zbl 0802.12005), F.Ulmer and J. A. Weil, J. Symb. Comput. 22, 179-200 (1996; Zbl 0871.12008)] using invariant theory. Applying these conditions and the differential invariants of a differential equation we deduce an alternative method to the algorithms given in (loc. cit.) for computing Liouvillian solutions. For irreducible second order equations our method determines solutions by formulas in all but three cases." } \end{chunk} \index{Jacquemard, Alain} \index{Khechichine-Mourtada, F.Z.} \index{Mourtada, A.} \begin{chunk}{axiom.bib} @article{Jacq97, author = "Jacquemard, Alain and Khechichine-Mourtada, F.Z. and Mourtada, A.", title = "Formal algorithms applied to the study of the cyclicity of a generic algebraic polycycle with four hyperbolic crests", journal = "Nonlinearity", volume = "10", number = "1", pages = "19-53", year = "1997", keywords = "axiomref", comment = "french", abstract = "Drawing on the work of Mourtada, we show that a family of vector fields with a generic algebraic polycycle of four hyperbolic apices possesses a maximum capacity of four limit cycles. This cyclicity is attained in an opening connecting the parameters which the edge contains, in particular a generic line of singularities of dovetail type. We also give an asymptotic estimation of the volume of this opening, as well as an explicit example of a family of polynomial vector fields replicating the above-described conditions and possessing five limit cycles. The methods employed are very diverse: geometrical arguments (Thom’s theory of catastrophes and the theory of algebraic singularities), developments from Puiseux, the number of major roots by Descartes’ law and calculated exactly by Sturm series, and other specific methods for formal calculus, such as for example the cylindrical algebraic decomposition and the resolution of algebraic systems via the construction of Gröbner bases. The calculations have been executed formally, that is to say without making the least appeal to numerical approximation, in using the formal calculus system AXIOM." } \end{chunk} \index{Lambe, Larry A.} \index{Radford, David E.} \begin{chunk}{axiom.bib} @book{Lamb97, author = "Lambe, Larry A. and Radford, David E.", title = "Introduction to the quantum Yang-Baxter equation and quantum groups: an algebraic approach", booktitle = "Mathematics and its Applications", publisher = "Kluwer Adademic Publishers", year = "1997", keywords = "axiomref", abstract = "The quantum Yang-Baxter equation (QYBE) has roots in statistical mechanics and the inverse scattering method and leads to a natural construction of a bialgebra. It turns out to have important connections with knot theory and invariants of 3-manifolds. There are now available many reference books to quantum groups and these various applications. The book under review develops the algebraic underpinning and theory of the QYBE, including the constant form and the one and two parameter forms. We give a brief description of the chapters. Chapter 1 (together with an Appendix) gives the algebraic preliminaries involving coalgebras, bialgebras, Hopf algebras, modules and comodules. Chapter 2 introduces the various forms of the QYBE, and the basic algebraic structures associated to them, including Faddeev-Reshetikhin-Takhtadzhan (FRT) construction. Chapter 3 explores various categorical settings for the constant form of the QYBE, the most basic being the category of left QYB modules over a bialgebra and the notion of algebras, coalgebras, etc. in this category. Chapter 4 develops universal mapping properties of the FRT construction and its reduced version, and the authors investigate when the reduced FRT construction leads to a pointed bialgebra or a pointed Hopf algebra. Chapter 5 develops the quantum groups associated to SL(2), i.e., the quantum universal enveloping algebra, and the quantum function algebra. Chapter 6 introduces quasitriangular Hopf algebras, and discusses how the finite-dimensional ones give rise to solutions of the QYBE through their representation theory. The most important example is the Drinfeld double of a finite-dimensional Hopf algebra. The authors note (through an exercise!) that every finite-dimensional Hopf algebra is the reduced FRT construction of some solution to the QYBE. Chapter 7 introduces coquasitriangular bialgebras, the most important being the FRT and the reduced FRT constructions. There are some generalizations here to the one-parameter form of the QYBE. Chapter 8 uses all the previously developed techniques to find solutions of the QYBE in certain cases, including the one-parameter form. Some of these were discovered by computer algebra methods. The final chapter 9 gives a brief discussion of certain categorical constructions and the QYBE is certain fairly abstract categories, motivated by the fact that the FRT construction is a coend. This book fills an important niche in the literature involving the QYBE by highlighting the algebraic aspects and applications. Although this is basically a reference book, it includes so many important parts of the study of Hopf algebras that it could be used as a textbook for a certain type of course on Hopf algebras and quantum groups, and certainly as supplementary reading material for such a course. There are frequent exercises which would be useful for such purposes. Besides being a basic source book, the authors include some new results and some novel approaches to earlier results. All this makes this book a most welcome addition to the quantum group literature." } \end{chunk} \index{Letichevskij, A. Alexander} \index{Marinchenko, V. G.} \begin{chunk}{axiom.bib} @article{Leti97, author = "Letichevskij, A. Alexander and Marinchenko, V. G.", title = "Objects in algebraic programming system", journal = "Cybern. Syst. Anal.", volume = "33", number = "2", pages = "283-299", year = "1997", keywords = "axiomref", comment = "translated from Russian", abstract = "The algebraic programming system (APS) developed at the V. M. Glushkov Institute of Cybernetics of the Academy of Sciences of the Ukrainian SSR integrates the basic programming paradigms, including procedural, functional, algebraic, and logic programming. Algebraic programming in APS relies on special data structures, the so-called graph terms, which permit using diverse data and knowledge representations in relevant application domains. In the language APLAN, graph terms are described by expressions or systems of expressions of a many-sorted algebra of data. They may represent both objects of the application domain and reasoning about these objects. The option of setting an arbitrary interpretation of the operations in the algebra of data makes it possible to use APS as a basis for various extensions. Symbolic computation systems such as Scratchpad/AXIOM have acquired special importance. They provide various possibilities of manipulating typed mathematical objects, including objects of complex hierarchical structure. This is a natural requirement when working with algebraic objects. In particular, the properties of many algebraic structures (such as groups, rings, fields, etc.) are naturally hierarchical-modular. The Institute of Cybernetics and the Kherson Teachers’ College have developed an instruction-oriented computer algebra system AIST. The AIST kernel is a hierarchical structure of mathematical concepts described in the APS language. However, construction of new applications on the basis of this hierarchical structure has proved difficult. The system kernel can be made more flexible by providing tools for flexible description of hierarchical structures of mathematical concepts. In this article, we describe an extension of the language APLAN, which provides tools for the object-oriented style of programming. This is one of the possible ways of introducing types in APS. The object-oriented technology also can be used to develop a hierarchical system of mathematical objects." } \end{chunk} \index{Schwarzweller, Christoph} \begin{chunk}{axiom.bib} @phdthesis{Schw97, author = "Schwarzweller, Christoph", title = "MIZAR verification of generic algebraic algorithms", school = "University of Tubingen", year = "1997", paper = "Schw97.pdf", keywords = "axiomref", abstract = "Although generic programming founds more and more attention – nowadays generic programming languages as well as generic libraries exist – there are hardly approaches for the verification of generic algorithms or generic libraries. This thesis deals with generic algorithms in the field of computer algebra. We propose the Mizar system as a theorem prover capable of verifying generic algorithms on an appropriate abstract level. The main advantage of the MIZAR theorem prover is its special input language that enables textbook style presentation of proofs. For generic versions of Brown/Henrici addition and of Euclidean’s algorithm we give complete correctness proofs written in the MIZAR language. Moreover, we do not only prove algorithms correct in the usual sense. In addition we show how to check, using the MIZAR system, that a generic algebraic algorithm is correctly instantiated with a particular domain. Answering this question that especially arises if one wants to implement generic programming languages, in the field of computer algebra requires nontrivial mathematical knowledge. To build a verification system using the MIZAR theorem prover, we also implemented a generator which almost automatically computes for a given algorithm a set of theorems that imply the correctness of this algorithm." } \end{chunk} \index{Zenger, Christoph} \begin{chunk}{axiom.bib} @article{Zeng97, article = "Zenger, Christoph", title = "Indexed types", journal = "Theor. Comput. Sci.", volume = "187", numbers = "1-2", pages = "147-165", year = "1997", keywords = "axiomref", paper = "Zeng97.pdf", abstract = "A new extension of the Hindley/Milner type system is proposed. The type system has algebraic types, that have not only type parameters but also value parameters (indices). This allows for example to parameterize matrices and vectors by their size and to check size compatibility statically. This is especially of interest in computer algebra." } \end{chunk} \index{Bernardin, Laurent} \begin{chunk}{axiom.bib} @article{Bern96, author = "Benardin, Laurent", title = "A review of symbolic solvers", journal = "SIGSAM Bull.", volume = "30", number = "1", pages = "9-20", year = "1996", keywords = "axiomref", paper = "Bern96.pdf", abstract = "Solving equations and systems of equations symbolically is a key feature of every Computer Algebra System. This review examines the capabilities of the six best known general purpose systems to date in the area of general algebraic and transcendental equation solving. Areas explicitly not covered by this review are differential equations and numeric or polynomial system solving as special purpose systems exist for these kinds of problems. The aim is to provide a benchmark for comparing Computer Algebra Systems in a specific domain. We do not intend to give a rating of overall capabilities as for example in [9]. 1 The Contestants We compare six major Computer Algebra Systems. Axiom 2.0 [7], Derive 3.06 [1], Macsyma 420 [8], Maple V R4 [3], Mathematica 2.2 [10], MuPAD 1.2.9 [5] and Reduce 3.6 [6]. When available, we tried to use the latest shipping version of each system. 2 The Problem Set The following table presents the set of 80 problems that we used to evaluate the different solvers..." } \end{chunk} \index{Wester, Michael J.} \begin{chunk}{axiom.bib} @misc{Westxx, author = "Wester, Michael J.", title = "Computer Algebra Synonyms", keywords = "axiomref", url = "http://math.unm.edu/~wester/cas/synonyms.pdf", paper = "Westxx.pdf", abstract = "The following is a collection of synonyms for various operations in the seven general purpose computer algebra systems {\bf Axiom}, {\bf Derive}, {\bf Macsyma}, {\bf Maple}, {\bf Mathematica}, {\bf MuPAD}, and {\bf Reduce}. This collection does not attempt to be comprehensive, but hopefully it will be useful in giving an indication of how to translate between the syntaxes used by the different systems in many common situations. Note that for a blank entry means that there is no exact translation of a particular operation for the indicated system, but it may still be possible to work around this lack with a related functionality." } \end{chunk} \index{Wester, Michael J.} \begin{chunk}{axiom.bib} @misc{West95, author = "Wester, Michael J.", title = "A Review of CAS Mathematical Capabilities", year = "1995", keywords = "axiomref", paper = "West95.pdf", url = "http://math.unm.edu/~wester/cas/Paper.ps", abstract = "Computer algebra systems (CASs) have become an important computational tool in the last decade. General purpose CASs, which are designed to solve a wide variety of problems, have gained special prominance. In this paper, the capabilities of seven major general purpose CASs (Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, and Reduce) are reviewed on 131 short problems covering a broad range of (primarily) symbolic mathematics. A demo was developed for each CAS, run and the results evaluated. Problems were graded in terms of whether it was easy or difficult or possible to produce an answer and if an answer was produced, whether it was correct. It is the author's hope that this review will encourage the development of a comprehensive CAS test suite." } \end{chunk} \index{Apel, Joachim} \index{Klaus, Uwe} \begin{chunk}{axiom.bib} @misc{Apel94, author = "Apel, Joachim and Klaus, Uwe", title = "Representing Polynomials in Computer Algebra Systems", year = "1994", paper = "Apel94.pdf", abstract = "There are discussed implementational aspects of the special-purpose computer algebra system FELIX designed for computations in constructive algebra. In particular, data types developed for the representation of and computation with commutative and non-commuative polynomials are described. Furthermore, comparison of time and memory requirements of different polynomial representations are reported." } \end{chunk} \index{Stoutemyer, David R.} \begin{chunk}{axiom.bib} @article{Stou91, author = "Stoutemyer, David R.", title = "Crimes and misdemeanors in the computer algebra trade", journal = "Notices of the American Mathematical Society", volume = "38", number = "7", pages = "778-785", year = "1991" } \end{chunk} \index{Sangwin, Chris} \begin{chunk}{axiom.bib} @misc{Sang10, author = "Sangwin, Chris", title = "Intriguing Integrals: Part I and II", year = "2010", url1 = "https://plus.maths.org/issue54/features/sangwin/2pdf/index.html/op.pdf", paper1 = "Sang10a.pdf", url2 = "https://plus.maths.org/issue54/features/sangwin2/2pdf/index.html/op.pdf", paper2 = "Sang10b.pdf" } \end{chunk} \index{Evans, Brian} \begin{chunk}{axiom.bib} @misc{Evanxx, author = "Evans, Brian", title = "History of CA Systems", url = "http://felix.unife.it/Root/d-Mathematics/d-The-mathematician/d-History-of-mathematics/t-History-of-computer-algebra", paper = "Evanxx.txt" } \end{chunk} \index{Martin, Ursula} \index{Shand, D.} \begin{chunk}{axiom.bib} @misc{Mart97, author = "Martin, Ursula and Shand, D", title = "Investigating some Embedded Verification Techniques for Computer Algebra Systems", url = "http://www.risc.jku.at/conferences/Theorema/papers/shand.ps.gz", paper = "Mart97.ps", abstract = " This paper reports some preliminary ideas on a collaborative project between St. Andrews University in the UK and NAG Ltd. The project aims to use embedded verification techniques to improve the reliability and mathematical soundness of computer algebra systems. We give some history of attempts to integrate computer algebra systems and automated theorem provers and discuss possible advantages and disadvantages of these approaches. We also discuss some possible case studies." } \end{chunk} \index{Tonisson, Eno} \begin{chunk}{axiom.bib} @article{Tonixx, author = "Tonisson, Eno", title = "Branch Completeness in School Mathematics and in Computer Algebra Systems", journal = "The Electronic Journal of Mathematics and Technology", volume = "1", number = "1", issn = "1933-2823", paper = "Tonixx.pdf", url = "https://php.radford.edu/~ejmt/deliveryBoy.php?paper=eJMT_v1n3p5", abstract = "In many cases when solving school algebra problems (e.g. simplifying an expression, solving an equation), the solution is separable into branches in some manner. The paper describes some approaches to branches that are used in school textbooks and computer algebra systems and compares them with mathematically branch-complete solutions. It tries to identify possible reasons behind different approaches and also indicate some ideas how such differences could be explained to the students." } \end{chunk} \index{Beeson, Michael} \begin{chunk}{axiom.bib} @misc{Beesxx, author = "Beeson, Michael", title = "Automatic Generation of Epsilon-Delta Proofs of Continuity", url = "http://www.michaelbeeson.com/research/papers/aisc.pdf", paper = "Beesxx.pdf", abstract = "As part of a project on automatic generation of proofs involving both logic and computation, we have automated the production of some proofs involving epsilon-delta arguments. These proofs involve two or three quantifiers on the logical side, and on the computational side, they involve algebra, trigonometry, and some calculus. At the border of logic and computation, they involve several types of arguments involving inequalities, including transitivity chaining and several types of bounding arguments, in which bounds are sought that do not depend on certain variables. Control mechanisms have been developed for intermixing logical deduction steps with computational steps and with inequality reasoning. Problems discussed here as examples involve the continuity and uniform continuity of various specific functions." } \end{chunk} \index{Ballarin, Clemens} \index{Paulson, Lawrence C.} \begin{chunk} @misc{Ball98, author = "Ballarin, Clemens and Paulson, Lawrence C.", title = "Reasoning about Coding Theory: The Benefits We Get from Computer Algebra", year = "1998", url = http://www21.in.tum.de/~ballarin/publications/aisc98.pdf", paper = "Ball98.pdf", abstract = "The use of computer algebra is usually considered beneficial for mechanised reasoning in mathematical domains. We present a case study, in the application domain of coding theory, that supports this claim: the mechanised proof depends on non-trivial algorithms from computer algebra and increase the reasoning power of the theorem prover. The unsoundness of computer algebra systems is a major problem in interfacing them to theorem provers. Our approach to obtaining a sound overall system is not blanket distrust but based on the distinction between algorithms we call sound and {\sl ad hoc} respectively. This distinction is blurred in most computer algebra systems OUr experimental interface therefore uses a computer algebra library. It is based on theorem templates, which provide formal specifications for the algorithms." } \end{chunk} \index{Aslaksen, Helmer} \begin{chunk}{axiom.bib} @article{Asla96, author = "Aslaksen, Helmer", title = "Multiple-valued complex functions and computer algebra", journal = "SIGSAM Bulletin", volume = "30", number = "2", year = "1996", pages = "12-20", paper = "Asla96.pdf", url = "http://www.math.nus.edu.sg/aslaksen/papers/cacas.pdf", abstract = "I recently taught a course on complex analysis. That forced me to think more carefully about branches. Being interested in computer algebra, it was only natural that I wanted to see how such programs dealt with these problems. I was also inspired by a paper by Stoutemyer. While programs like Derive, Maple, Mathematica and Reduce are very powerful, they also have their fair share of problems. In particular, branches are somewhat of an Achilles' heel for them. As is well-known, the complex logarithm function is properly defined as a multiple-valued function. And since the general power and exponential functions are defined in terms of the logarithm function, they are also multiple-valued. But for actual computations, we need to make them single valued, which we do by choosing a branch. In Section 2, we will consider some transformation rules for branches of multiple-valued complex functions in painstaking detail. The purpose of this short article is not to do a comprehensive comparative study of different computer algebra systems. My goal is simply to make the readers aware of some of the problems, and to encourage the readers to sit down and experiment with their favourite programs." } \end{chunk} \index{Fateman, Richard J.} \begin{chunk}{axiom.bib} @InProceedings{Fate96, author = "Fateman, Richard J.", title = "A Review of Symbolic Solvers", booktitle = "Proc 1996 ISSAC", series = "ISSAC 96", year = "1996", pages = "86-94", keywords = "axiomref", keywords = "axiomref", paper = "Fate96.pdf", url = "http://http.cs.berkeley.edu/~fateman/papers/eval.ps", abstract = "Evaluation'' of expressions and programs in a computer algebra system is central to every system, but inevitably fails to provide complete satisfaction. Here we explain the conflicting requirements, describe some solutions from current systems, and propose alternatives that might be preferable sometimes. We give examples primarily from Axiom, Macsyma, Maple, Mathematica, with passing metion of a few other systems." } \end{chunk} \index{Fateman, Richard J.} \begin{chunk}{axiom.bib} @misc{Fate05, author = "Fateman, Richard J.", title = "An incremental approach to building a mathematical expert out of software", conference = "Axiom Computer Algebra Conference", location = "City College of New York, CAISS project", year = "2005", month = "April", day = "19", url = "http://www.cs.berkeley.edu/~fateman/papers/axiom.pdf", paper = "Fat05.pdf", keywords = "axiomref" } \end{chunk} \index{Gr\"abe, Hans-Gert} \begin{chunk}{axiom.bib} @misc{Grab98, author = "Grabe, Hans-Gert", title = "About the Polynomial System Solve Facility of Axiom, Macsyma, Maple Mathematica, MuPAD, and Reduce", paper = "Grab98.pdf", url = "https://www.informatik.uni-leipzig.de/~graebe/ComputerAlgebra/Publications/WesterBook.pdf", keywords = "axiomref", abstract = "We report on some experiences with the general purpose Computer Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and Reduce solving systems of polynomial equations and the way they present their solutions. This snapshot (taken in the spring of 1996) of the current power of the different systems in a special area concentrates on both CPU-times and the quality of the output." } \end{chunk} \index{Gr\"abe, Hans-Gert} \begin{chunk}{axiom.bib} @misc{Grab06, author = "Grabe, Hans-Gert", title = "The Groebner Factorizer and Polynomial System Solving", year = "2006", keywords = "axiomref", report = "Special Semester on Groebner Bases", location = "Linz", paper = "Grab06.pdf", url = "https://www.ricam.oeaw.ac.at/specsem/srs/groeb/download/06\_02\_Solver.pdf", abstract = "Let S := k[x_1,\ldots, x_n] be the polynomial ring in the variables x_1,\ldots,x_n over the field k and B := \{f_1,\ldots,f_m\} \subset S be a finite system of polynomials. Denote by I(B) the ideal generated by these polynomials. One of the major tasks of constructive commutative algebra is the derivation of information about the structure of \[V(B):=\{a \in K^n : \forall f \in B{\rm\ such\ that\ }f(a)=0\}$ the set of common zeroes of the system$B$over an algebraically closed extension$K$of$k$. Splitting the system into smaller ones, solving them separately, and patching all solutions together is often a good guess for a quick solution of even highly nontrivial problems. This can be done by several techniques, e.g., characteristic sets, resultants, the Groebner factorizer or some ad hoc methods. Of course, such a strategy makes sense only for problems that really will split, i.e., for reducible varieties of solutions. Surprisingly often, problems coming from 11real life'' fulfill this condition. Among the methods to split polynomial systems into smaller pieces probably the Groebner factor- izer method attracted the most theoretical attention, see Czapor ([4, 5]), Davenport ([6]), Melenk, M ̈oller and Neun ([16, 17]) and Gr ̈abe ([13, 14]). General purpose Computer Algebra Systems (CAS) are well suited for such an approach, since they make available both a (more or less) well tuned implementation of the classical Groebner algorithm and an effective multivariate polynomial factorizer. Furthermore it turned out that the Groebner factorizer is not only a good heuristic approach for splitting, but its output is also usually a collection of almost prime components. Their description allows a much deeper understanding of the structure of the set of zeroes compared to the result of a sole Groebner basis computation. Of course, for special purposes a general CAS as a multipurpose mathematical assistant can’t offer the same power as specialized software with efficiently implemented and well adapted algorithms and data types. For polynomial system solving, such specialized software has to implement two algorithmically complex tasks, solving and splitting, and until recently none of the specialized systems (as e.g., GB, Macaulay, Singular, CoCoA, etc.) did both efficiently. Meanwhile, being very efficient computing (classical) Groebner bases, development efforts are also directed, not only for performance reasons, towards a better inclusion of factorization into such specialized systems. Needless to remark that it needs some skill to force a special system to answer questions and the user will probably first try his home system'' for an answer. Thus the polynomial systems solving facility of the different CAS should behave especially well on such polynomial systems that are hard enough not to be done by hand, but not really hard to require special efforts. It should invoke a convenient interface to get the solutions in a form that is (correct and) well suited for further analysis in the familiar environment of the given CAS as the personal mathematical assistant." } \end{chunk} \index{Corless, Robert M.} \index{Jeffrey, David J.} \index{Watt, Stephen M.} \index{Bradford, Russell} \index{Davenport, James H.} \begin{chunk}{axiom.bib} @misc{Corl0, author = "Corless, Robert M. and Jeffrey, David J. and Watt, Stephen M. and Bradford, Russell and Davenport, James H.", title = "Reasoning about the elementary functions of complex analysis", url = "http://www.csd.uwo.ca/~watt/pub/reprints/2002-amai-reasoning.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 complex-variable analysis." } \end{chunk} \index{Touratier, Emmanuel} \begin{chunk}{axiom.bib} @misc{Tour98, author = "Touratier, Emmanuel", title = {Etude du typage dans le syst\eme de calcul scientifique Aldor}, comment = "Study of types in the Aldor scientific computation system", year = "1998", paper = "Tour98.pdf", url = "http://axiom-wiki.newsynthesis.org/public/refs/Aldor-T1998_04.pdf", keywords = "axiomref" } \end{chunk} \index{Seiler, Werner Markus} \begin{chunk}{axiom.bib} @misc{Seil95, author = "Seiler, Werner Markus", title = "Applying AXIOM to partial differential equations", institution = {Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik}, year = "1995", type = "Internal Report", number = "95-17", url = "http://axiom-wiki.newsynthesis.org/public/refs/Axiom-pdf.pdf", paper = "Seil95.pdf", keywords = "axiomref", abstract = "We present an Axiom environment called JET for geometric computations with partial differential equations within the framework of the jet bundle formalism. This comprises expecially the completion of a given differential equation to an involutive one according to the Cartan-Kuranishi Theorem and the setting up of the determining system for the generators of classical and non-classical Lie symmetries. Details of the implementations are described and applications are given. An appendix contains tables of all exported functions." } \end{chunk} \index{Davenport, James H.} \begin{chunk}{axiom.bib} @misc{Dave84a, author = "Davenport, James H.", title = "A New Algebra System", paper = "Dave84a.pdf", keywords = "axiomref", url = "http://axiom-wiki.newsynthesis.org/public/refs/Davenport-1984-a\_new\_algebra\_system.pdf", abstract = "Seminal internal paper discussing Axiom design decisions." } \end{chunk} \index{Conrad, Marc} \index{French, Tim} \index{Maple, Carsten} \index{Pott, Sandra} \begin{chunk}{axiom.bib} @misc{Conrxxa, author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra", title = "Approaching Inheritance from a Natural Mathematical Perspective and from a Java Driven Viewpoint: a Comparative Review", keywords = "axiomref", url = "http://axiom-wiki.newsynthesis.org/public/refs/McTfCmSp-axiom.pdf", paper = "Conrxxa.pdf", abstract = " It is well-known that few object-oriented programming languages allow objects to change their nature at run-time. 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 object-oriented programming languages to reflect the dynamic nature of problems, particularly those arising in a mathematical context. It is from this context that we present a framework that realistically represents the dynamic and evolving characteristic of problems and algorithms." } \end{chunk} \index{Meijer, Erik} \index{Fokkinga, Maarten} \index{Paterson, Ross} \begin{chunk}{axiom.bib} @misc{Meij91, author = "Meijer, Erik and Fokkinga, Maarten and Paterson, Ross", title = "Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire", url = "http://eprints.eemcs.utwente.nl/7281/01/db-utwente-40501F46.pdf", paper = "Meij91.pdf", abstract = " We develop a calculus for lazy functional programming based on recursion operators associated with data type definitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example functions in Bird and Wadler's Introduction to Functional Programming'' can be expressed using these operators." } \end{chunk} \index{Robidoux, Nicolas} \begin{chunk}{axiom.bib} @misc{Robi93, author = "Robidoux, Nicolas", title = "Does Axiom Solve Systems of O.D.E's Like Mathematica?", year = "1993", paper = "Robi93.pdf", url = "http://axiom-wiki.newsynthesis.org/public/refs/Robidoux.pdf", keywords = "axiomref", abstract = " If I were demonstrating Axiom and were asked this question, my reply would be No, but I am not sure that this is a bad thing''. And I would illustrate this with the following example. Consider the following system of O.D.E.'s $\begin{array}{rcl} \frac{dx_1}{dt} & = & \left(1+\frac{cos t}{2+sin t}\right)x_1\\ \frac{dx_2}{dt} & = & x_1 - x_2 \end{array}$ This is a very simple system:$x_1$is actually uncoupled from$x_2$" } \end{chunk} \index{Davenport, James H.} \index{Faure, Christ\'ele} \begin{chunk}{axiom.bib} @misc{Davexx, author = {Davenport, James; Faure, Christ\'ele}, title = "The Unknown in Computer Algebra", url = "http://axiom-wiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf", paper = "Davexx.pdf", keywords = "axiomref", abstract = " Computer algebra systems have to deal with the confusion between programming variables'' and mathematical symbols''. We claim that they should also deal with unknowns'', i.e. elements whose values are unknown, but whose type is known. For examples$x^p \ne x$if$x$is a symbol, but$x^p = x$if$x \in GF(p)$. We show how we have extended Axiom to deal with this concept." } \end{chunk} \index{Davenport, James H.} \begin{chunk}{axiom.bib} @techreport{Dave92b, author = "Davenport, James H.", title = "How does one program in the AXIOM system?", institution = "Numerical Algorithms Group, Inc.", year = "1992", type = "technical report", number = "TR6/92 (ATR/4)(NP2493)", url = "http://www.nag.co.uk/doc/TechRep/axiomtr.html", paper = "Dave92b.pdf", keywords = "axiomref", abstract = "Axiom is a computer algebra system superficially like many others, but fundamentally different in its internal construction, and therefore in the possibilities it offers to its users and programmers. In these lecture notes, we will explain, by example, the methodology that the author uses for programming substantial bits of mathematics in Axiom." } \end{chunk} \index{Youssef, Saul} \begin{chunk}{axiom.bib} @misc{Yous04, author = "Youssef, Saul", title = "Prospects for Category Theory in Aldor", year = "2004", url = "http://axiom-wiki.newsynthesis.org/public/refs/Youssef-ProspectsForCategoryTheoryInAldor.pdf", paper = "Yous04.pdf", abstract = "Ways of encorporating category theory constructions and results into the Aldor language are discussed. The main features of Aldor which make this possible are identified, examples of categorical constructions are provided and a suggestion is made for a foundation for rigorous results." } \end{chunk} \index{Carpent, Quentin} \index{Conil, Christophe} \begin{chunk}{axiom.bib} @misc{Carp04, author = "Carpent, Quentin and Conil, Christophe", title = "Utilisation de logiciels libres pour la r\'ealisation de TP MT26", year = "2004", paper = "Carp04.pdf", url = "http://axiom-wiki.newsynthesis.org/public/refs/ac20.pdf", keywords = "axiomref", comment = "french", abstract = "radicalSolve(x**3+x**2-7=0,x)" } \end{chunk} \index{Naylor, William A.} \index{Padget, Julian} \begin{chunk}{axiom.bib} @InProceedings{Nayl06, author = "Naylor, William and Padget, Julian", title = "From Untyped to Polymorphically Typed Objects in Mathematical Web Services", paper = "NPxx.pdf", series = Lecture Notes in Computer Science", volume = "4108", pages = "222-236", year = "2006", keywords = "axiomref", abstract = "OpenMath is a widely recognized approach to the semantic markup of mathematics that is often used for communication between OpenMath compliant systems. The Aldor language has a sophisticated category-based type system that was specifically developed for the purpose of modelling mathematical structures, while the system itself supports the creation of small-footprint applications suitable for deployment as web services. In this paper we present our first results of how one may perform translations from generic OpenMath objects into values in specific Aldor domains, describing how the Aldor interfae domain ExpresstionTree is used to achieve this. We outline our Aldor implementation of an OpenMath translator, and describe an efficient extention of this to the Parser category. In addition, the Aldor service creation and invocation mechanism are explained. Thus we are in a position to develop and deploy mathematical web services whose descriptions may be directly derived from Aldor's rich type language." } \end{chunk} \index{Watt, Stephen M.} \index{Broadbery, Peter A.} \index{Dooley, Sam} \index{Iglio, Pietro} \begin{chunk}{axiom.bib} @techreport{Watt94, author = "Watt, Stephen M. and Broadbery, Peter A. and Dooley, Samuel S. and Iglio, Pietro", title = "A First Report on the A\# Compiler (including benchmarks)", institution = "IBM Research", year = "1994", type = "technical report", number = "RC19529 (85075)", paper = "Watt94.pdf", url = "http://axiom-wiki.newsynthesis.org/public/refs/axiom-aldor-a-sharp.pdf", keywords = "axiomref", abstract = "The$A^{#}$compiler allows users of computer algebra to develop programs in a context where multiple programming languages are employed. The compiler translates programs written in the$A^{#}$programming language to a low-level intermediate language, Foam, from which it can generate stand-alone programs, native object libraries to be linked with other applications, or code to be read into closed environments. In addition, Foam code may be directly executed using an interpreter provided with the$A^{#}$compiler. The$A^{#}$programming language provides support for object-oriented and functional programming styles. It is higher-order'' in the sense that both types and functions are first class, and may be manipulated in the same ways as any other values. The primary considerations in the formulation of the language have been generality, composibility, and efficiency. The language has been designed to admit a number of important optimizations, allowing compilation to machine code which is in many instances of efficiency comparable to that produced by a C or Fortran compiler. The original motivation for$A^{#}$comes from the field of computer algebra: to provide an improved extension language for the Axiom computer algebra system." } \end{chunk} \index{Lambe, Larry A.} \index{Luczak, Richard} \begin{chunk}{axiom.bib} @article{Lamb93a, article = "Lambe, Larry and Luczak, Richard", title = "Object-Oriented Mathematical Programming and Symbolic/Numeric Interface", journal = "3rd Int. Conf. on Expert Systems in Numerical Computing", year = "1993", url = "http://axiom-wiki.newsynthesis.org/public/refs/axiom-fem.pdf", paper = "Lamb93a.pdf", keywords = "axiomref", abstract = "The Axiom language is based on the notions of categories'', domains'', and packages''. These concepts are used to build an interface between symbolic and numeric calculations. In particular, an interface to the NAG Fortran Library and Axiom's algebra and graphics facilities is presented. Some examples of numerical calculations in a symbolic computational environment are also included using the finite element method. While the examples are elementary, we believe that they point to very powerful methods for combining numeric and symbolic computational techniques." } \end{chunk} \index{Griesmer, James H.} \index{Jenks, Richard D.} \begin{chunk}{axiom.bib} @InProceedings{Grie71, author = "Griesmer, James H. and Jenks, Richard D.", title = "SCRATCHPAD/1 -- an interactive facility for symbolic mathematics", booktitle = "Proc. second ACM Symposium on Symbolic and Algebraic Manipulation", series = "SYMSAC 71", year = "1971", pages = "42-58", url = "http://delivery.acm.org/10.1145/810000/806266/p42-griesmer.pdf", paper = "GJ71.pdf", keywords = "axiomref", abstract = " The SCRATCHPAD/1 system is designed to provide an interactive symbolic computational facility for the mathematician user. The system features a user language designed to capture the style and succinctness of mathematical notation, together with a facility for conveniently introducing new notations into the language. A comprehensive system library incorporates symbolic capabilities provided by such systems as SIN, MATHLAB, and REDUCE." } \end{chunk} \index{Seiler, Werner Markus} \index{Calmet, J.} \begin{chunk}{axiom.bib} @misc{Seil95a, author = "Seiler, Werner Markus and Calmet, J.", title = "JET -- An Axiom Environment for Geometric Computations with Differential Equations", paper = "Seil95a.pdf", url = "http://axiom-wiki.newsynthesis.org/public/refs/axiom-jet95.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 Cartan-Kuranishi theorem and for setting up the determining equations for classical and non-classical point symmetries." } \end{chunk} committed Jun 29, 2016 Commits on Jun 28, 2016 1. Goal: Axiom build Somewhere along the way I fat-fingered a character delete causing the build to break. Sigh. committed Jun 28, 2016 2. Goal: Axiom Literate Programming \index{Salem, Fatima Khaled Abu} \begin{chunk}{axiom.bib} @phdthesis{Sale04, author = "Salem, Fatima Khaled Abu", title = "Factorisation Algorithms for Univariate and Bivariate Polynomials over Finite Fields", school = "Meron College", year = "2004", paper = "Sale04", url = "http://www.cs.aub.edu.lb/fa21/Dissertations/My\_thesis.pdf", abstract = "In this thesis we address algorithms for polynomial factorisation over finite fields. In the univariate case, we study a recent algorithm due to Niederreiter where the factorisation problem is reduced to solving a linear system over the finite field in question, and the solutions are used to produce the complete factorisation of the polynomials into irreducibles. We develop a new algorithm for solving the linear system using sparse Gaussian elimination with the Markowitz ordering strategy, and conjecture that the Niederreiter linear system is not only initially sparse, but also preserves its sparsity throughout the Gaussian elimination phase. We develop a new bulk synchronous parallel (BSP) algorithm base on the approach of Gottfert for extracting the factors of a polynomial using a basis of the Niederreiter solution set of$\mathbb{F}_2$. We improve upon the complexity and performance of the original algorithm, and produce binary univariate factorisations of trinomials up to degree 400000. We present a new approach to multivariate polynomial factorisation which incorporates ideas from polyhedral geometry, and generalises Hensel lifting. The contribution is an algorithm for factoring bivariate polynomials via polytopes which is able to exploit to some extent the sparsity of polynomials. We further show that the polytope method can be made sensitive to the number of nonzero terms of the input polynomial. We describe a sparse adaptation of the polytope method over finite fields of prime order which requires fewer bit operations and memory references for polynomials which are known to be the product of two sparse factors. Using this method, and to the best of our knowledge, we achieve a world record in binary bivariate factorisation of a sparse polynomial of degree 20000. We develop a BSP variant of the absolute irreducibility testing via polytopes given in [45], producing a more memory and run time efficient method that can provide wider ranges of applicability. We achieve absolute irreducibility testing of a bivariate and trivariate polynomial of degree 30000, and of multivariate polynomials with up to 3000 variables." } \end{chunk} \index{Gianni, P.} \index{Trager, B.} \begin{chunk}{axiom.bib} @article{Gian96, author = "Gianni, P. and Trager, B.", title = "Square-free algorithms in positive characteristic", journal = "J. of Applicable Algebra in Engineering, Communication and Computing", volume = "7", pages = "1-14", year = "1996", } \end{chunk} \index{Shoup, Victor} \begin{chunk}{axiom.bib} @InProceedings{Shou91, author = "Shoup, Victor", title = "A Fast Deterministic Algorithm for Factoring Polynomials over Finite Fields of Small Characteristic", booktitle = "Proc. ISSAC 1991", series = "ISSAC 1991", year = "1991", pages = "14-21", paper = "Shou91.pdf", url = "http://www.shoup.net/papers/quadfactor.pdf", abstract = "We present a new algorithm for factoring polynomials over finite fields. Our algorithm is deterministic, and its running time is almost'' quadratic when the characteristic is a small fixed prime. As such, our algorithm is asymptotically faster than previously known deterministic algorithms for factoring polynomials over finite fields of small characteristic." } \end{chunk} \index{von zur Gathen, Joachim} \index{Kaltofen, Erich} \begin{chunk}{axiom.bib} @Article{Gath85b, author = "{von zur Gathen}, Joachim and Kaltofen, E.", title = "Polynomial-Time Factorization of Multivariate Polynomials over Finite Fields", journal = "Math. Comput.", year = "1985", volume = "45", pages = "251-261", url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz", paper = "Gath85.ps", abstract = "We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e. in the degree of the polynomial and$log$(cardinality of field). The algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs. Also a deterministic version of the algorithm is discussed whose running time is polynomial in the degree of the input polynomial and the size of the field." } \end{chunk} \index{von zur Gathen, Joachim} \index{Panario, Daniel} \begin{chunk}{axiom.bib} @article{Gath01, author = "von zur Gathen, Joachim and Panario, Daniel", title = "Factoring Polynomials Over Finite Fields: A Survey", journal = "J. Symbolic Computation", year = "2001", volume = "31", pages = "3-17", paper = "Gath01.pdf", url = "http://people.csail.mit.edu/dmoshdov/courses/codes/poly-factorization.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 up-to-date bibliography of the problem. This paper gives algorithms for {\sl squarefree factorization}, {\sl distinct-degree factorization}, and {\sl equal-degree factorization}. The first and second algorithms are deterministic, the third is probabilistic." } \end{chunk} \index{Augot, Daniel} \index{Camion, Paul} \begin{chunk}{axiom.bib} @article{Augo97, author = "Augot, Daniel and Camion, Paul", title = "On the computation of minimal polynomials, cyclic vectors, and Frobenius forms", journal = "Linear Algebra Appl.", volume = "260", pages = "61-94", year = "1997", keywords = "axiomref", paper = "Augo97.pdf", abstract = "Algorithms related to the computation of the minimal polynomial of an$x\times n$matrix over a field$K$are introduced. The complexity of the first algorithm, where the complete factorization of the characteristic polynomial is needed, is$O(\sqrt{n}\cdot n^3)$. An iterative algorithm for finding the minimal polynomial has complexity$O(n^3+n^2m^2)$, where$m$is a parameter of the shift Hessenberg matrix used. The method does not require the knowlege of the characteristic polynomial. The average value of$m$is$O(log n)$. Next methods are discussed for finding a cyclic vector for a matrix. The authors first consider the case when its characteristic polynomial is squarefree. Using the shift Hessenberg form leads to an algorithm at cost$O(n^3 + n^2m^2)$. A more sophisticated recurrent procedure gives the result in$O(n^3)$steps. In particular, a normal basis for an extended finite field of size$q^n$will be obtained with complexity$O(n^3+n^2 log q)$. Finally, the Frobenius form is obtained with asymptotic average complexity$O(n^3 log n)$." } \end{chunk} \index{Bernardin, Laurent} \index{Monagan, Michael B.} \begin{chunk}{axiom.bib} @InProceedings{Bern97a, author = "Bernardin, Laurent and Monagan, Michael B.", title = "Efficient multivariate factorization over finite fields", booktitle = "Applied algebra, algebraic algorithms and error-correcting codes", series = "AAECC-12", year = "1997", location = "Toulouse, France", publisher = "Springer", pages = "15-28", keywords = "axiomref", paper = "Bern97a.pdf", url = "http://www.cecm.sfu.ca/~monaganm/papers/AAECC.pdf", abstract = "We describe the Maple implementation of multivariate factorization over general finite fields. Our first implementation is available in Maple V Release 3. We give selected details of the algorithms and show several ideas that were used to improve its efficiency. Most of the improvements presented here are incorporated in Maple V Release 4. In particular, we show that we needed a general tool for implementing computations in GF$(p^k)[x_1,x_2,\cdots,x_v]$. We also needed an efficient implementation of our algorithms$\mathbb{Z}_p[y][x]$in because any multivariate factorization may depend on several bivariate factorizations. The efficiency of our implementation is illustrated by the ability to factor bivariate polynomials with over a million monomials over a small prime field." } \end{chunk} \index{Bronstein, Manuel} \index{Weil, Jacques-Arthur} \begin{chunk}{axiom.bib} @article{Bron97a, author = "Bronstein, Manuel and Weil, Jacques-Arthur", title = "On Symmetric Powers of Differential Operators", series = "ISSAC'97", year = "1997", pages = "156-163", keywords = "axiomref", url = "http://www-sop.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} \index{Calmet, J.} \index{Campbell, J.A.} \begin{chunk}{axiom.bib} @article{Calm97, author = "Calmet, J. and Campbell, J.A.", title = "A perspective on symbolic mathematical computing and artificial intelligence", journal = "Ann. Math. Artif. Intell.", volume = "19", number = "3-4", pages = "261-277", year = "1997", keywords = "axiomref", paper = "Calm97.pdf", url = "http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.5425&rep=rep1&type=pdf", abstract = "The nature and history of the research area common to artificial intelligence and symbolic mathematical computation are examined, with particular reference to the topics having the greatest current amount of activity or potential for further development: mathematical knowledge-based computing environments, autonomous agents and multi-agent systems, transformation of problem descriptions in logics into algebraic forms, exploitation of machine learning, qualitative reasoning, and constraint-based programming. Knowledge representation, for mathematical knowledge, is identified as a central focus for much of this work. Several promising topics for further research are stated." } \end{chunk} committed Jun 28, 2016 3. Goal: Axiom Literate Programming \index{Rigal, Alain} \begin{chunk}{axiom.bib} @article{Riga99, author = "Rigal, Alain", title = "High-order compact schemes: Application to bidimensional unsteady diffusion-convection problems.", journal = "C. R. Acad. Sci.", volume = "328", number = "6", pages = "535-538", year = "1999", keywords = "axiomref", abstract = "For unsteady 2D diffusion-convection problems, we present two classes of compact difference schemes of order 2 in time and 4 in space. These finite difference schemes are essentially derived from 1D schemes, extensively analyzed in our previous paper [J. Comput. Phys. 114, No. 1, 59-76 (1994; Zbl 0807.65056)]. We propose two approaches: construction of 2D schemes as product of 1D schemes and global formulation of 2D schemes. Part II by M. Fournié [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 6, 539-542 (1999; reviewed below)] focuses on the development and analysis of global schemes with the assistance of symbolic computation software (AXIOM)." } \end{chunk} \index{Roesner, K. G.} \begin{chunk}{axiom.bib} @article{Roes99, author = "Roesner, K. G.", title = "Supersonic flow around accelerated and decelerated bodies, analysed by analytical methods", journal = "Z. Angew. Math. Mech.", volume = "79", number = "3", pages = "815-816", year = "1999", keywords = "axiomref", abstract = "By an extensive use of the computer algebra system AXIOM, a power series expansion with respect to the radial variable$r$is used to describe the accelerated or decelerated supersonic flow field around the tip of slender conical bodies. The set of coupled nonlinear differential equations for the coefficient functions, depending on$\theta$and$t$, is derived in closed form, and the first and second approximation of the coefficient functions are determined numerically." } \end{chunk} \index{Stroeker, Roelof J.} \index{Kaashoek, Johan F.} \begin{chunk}{axiom.bib} @book{Stro99, author = "Stroeker, Roelof J. and Kaashoek, Johan F.", title = "Discovering mathematics with Maple. An interactive exploration for mathematicians, engineers and econometricians", year = "1999", publisher = "Birkhauser", keywords = "axiomref", abstract = "During the past decade, the mathematical computer software packages such as Mathematica, Maple, MATLAB (Axiom, Derive, Macsyma, MuPad are some further examples of such software) [see Macsyma 2.3. Lite – the student edition (1998; Zbl 0911.68089); B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan, and S. M. Watt, Maple V Library reference manual (1991; Zbl 0763.68046); J. L. Zachary, Introduction to scientific programming. Computational problem solving using Mathematica and C (1997; Zbl 0891.68053); The student edition of MATLAB. Student user guide. The problem-solving tool for engineers, mathematicians, and scientists (1992; Zbl 0782.65001); H. Benker, Ingenieurmathematik mit Computeralgebra-Systemen. AXIOM, DERIVE, MACSYMA, MAPLE, MATHCAD, MATHEMATICA, MATLAB und MuPAD in der Anwendung (1998; Zbl 0909.68109); W. Koepf, Hohere Analysis mit DERIVE (1994; Zbl 0819.26003)] have greatly faciliated mathematical experiments and have thus become popular tools for the modern mathematician. It is a pity that most of these packages are quite expensive, and that the frequently upgraded versions are not free for the owners of the earlier versions (fortunately, there are inexpensive student versions of some of these packages). There is a constant demand of instructional textbooks by users of these packages. This demand is reflected in the growing number of such textbooks. Many of these books provide software support (diskette, CD-ROM, access by ftp). Such a textbook should meet, in my opinion, the following criteria: (1) The size should be small, not bulky like the complete technical descriptions of the software. (2) There should be a lot of examples of the use of the software covering a wide range of mathematical topics. Electronic versions of these examples should be made available for free to the users of the textbook (e.g. diskette/CD-ROM, access by ftp). (3) There should be a good supply of exercises covering the basic mathematical applications. (4) The book should be visually pleasing, easy to read, have good indexes and provide pointers to other books and electronic sources of information. The book under review provides, in addition to the actual text, an interactive exploratorium of its topics, based on the mechanism of Maple worksheets. These worksheets can be opened'' by the Maple program and they form a mixture of usual text, hypertext, and Maple commands and have a nice style appearance. They also can be exported'' in a file and included in a file for further treatment. The book meets all the aforementioned criteria (1)-(4) with elegance. There are many exercises which cover all the usual mathematical topics from linear algebra to differential equations and statistics. A valuable feature is an appendix with hints and answers for all exercises. One of the highlights of the book is the examination of Riemann's non-differentiable function $x \mapsto \sum_{k=1}^\infty{k^{-2}} sin(\pi kx)$ which is differentiable only at the rational points$p/q$with$p$and$q$odd and relatively prime, where its derivative is$-1/2$. The book is intended for students of mathematics, engineering sciences, and econometry. This book is an ideal guide for this purpose and it could probably be used along, without the bulky technical documentation of the Maple language. Note that Maple has a comprehensive on-line help program, which contains large parts of the original documentation." } \end{chunk} \index{Wester, Michael J.} \begin{chunk}{axiom.bib} @book{West99, author = "Wester, Michael J.", title = "Computer Algebra Systems. A practical guide", year = "1999", publisher = "Wiley", keywords = "axiomref", abstract = "In this book some of the most popular general purpose computer algebra systems (CAS), such as Mathematica, Maple, Derive, Axiom, MuPAD, and Macsyma, are examined. The strengths and weaknesses of these programs are compared and contrasted, and tutorial information for using these systems in various ways is given. The different packages are quantitatively compared using standard test suites, giving the possibility to asses the most appropriate for a particular user or application. The origins of these systems are revealed and many of their behaviors analyzed. This furnishes a feel for where the current computer algebra system state of the art stays and what can be expected for existing and future systems. The book is organized in several chapters written by different authors. Chapters 1,2, and 3 are organized as reviews, comparisons, and critiques of CAS capabilities. Then more technical issues are discussed considering different approaches taken by different CAS: simplifying square roots of square roots by denesting (chapter 4), complex number calculation (chapter 5), efficiently computing Chebyshev polynomials (chapter 6), solving single equations and systems of polynomial equations (chapters 7, 8), computing limits (chapter 9), multiple integration (chapter 10), solving ordinary differential equation (chapter 11), integration of nonlinear evolution equations (chapter 12), code generation (chapter 13), evaluation of expressions and programs in the embedded computer algebra programming language (chapter 14), and computer algebra in education (chapter 15). Chapter 16 covers the origin of CA, and, finally chapter 17 gives a list of most CAS available today." } \end{chunk} \index{Benker, Hans} \begin{chunk}{axiom.bib} @book{Benk98, author = "Benker, Hans", title = "Engineering mathematics with computer algebra systems", year = "1998", keywords = "axiomref", comment = "german" } \end{chunk} \index{Breuer, Thomas} \index{Linton, Steve} \begin{chunk}{axiom.bib} @InProceedings{Breu98, author = "Breuer, Thomas and Linton, Steve", title = "The GAP 4 type system organising algebraic algorithms", booktitle = "Proc. ISSAC 98", series = "ISSAC 98", year = "1998", publisher = "ACM Press", location = "Rostock, Germany", pages = "13-15", keywords = "axiomref", paper = "Breu98.pdf", url = "http://www.gap-system.org/Doc/Talks/paper.ps", abstract = "Version 4 of the GAP (Groups, Algorithms, Programming) system for computational discrete mathematics has a number of novel features. In this paper, we describe the type system, and the way in which it is used for method selection. This system is central to the organization of the library which is the main part of the GAP system. Unlike simpler object-oriented systems, GAP allows method selection based on the types of all arguments and on certain aspects of the relationship between the arguments. In addition, the type of an object can change, in a controlled way, during its life. This reflects information about the object which has been computed and stored. Individual methods can be written and installed independently. Furthermore, most checking of the arguments is done in a uniform way by the method selection system, making individual methods simpler and less prone to error. The methods are combined automatically to produce a powerful and usable system for interactive use or programming." } \end{chunk} \index{Linton, Stephen} \begin{chunk}{axiom.bib} @misc{Lint98, author = "Linton, Stephen", title = "The GAP 4 Type System Organising Algebraic Algorithms", paper = "Lint98.pdf", url = "http://www.gap-system.org/Doc/Talks/kobe.ps", keywords = "axiomref" } \end{chunk} \index{Diaz, Angel} \index{Kaltofen, Erich} \begin{chunk}{axiom.bib} @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", publisher = "ACM Press", year = "1998", pages = "30--37", url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/DiKa98.pdf", paper = "Diaz98.pdf", abstract = "The FOXBOX system puts in practice the black box representation of symbolic objects and provides algorithms for performing the symbolic calculus with such representations. Black box objects are stored as functions. For instance: a black box polynomial is a procedure that takes values for the variables as input and evaluates the polynomial at that given point. FOXBOX can compute the greatest common divisor and factorize polynomials in black box representation, producing as output new black boxes. It also can compute the standard sparse distributed representation of a black box polynomial, for example, one which was computed for an irreducible factor. We establish that the black box representation of objects can push the size of symbolic expressions far beyond what standard data structures could handle before. Furthermore, FOXBOX demonstrates the generic program design methodology. The FOXBOX system is written in C++. C++ template arguments provide for abstract domain types. Currently, FOXBOX can be compiled with SACLIB 1.1, Gnu-MP 1.0, and NTL 2.0 as its underlying field and polynomial arithmetic. Multiple arithmetic plugins can be used in the same computation. FOXBOX provides an MPI compliant distribution mechanism that allows for parallel and distributed execution of FOXBOX programs. Finally, FOXBOX plugs into a server/client-style Maple application interface." } \end{chunk} \index{Dooley, Samuel S.} \begin{chunk}{axiom.bib} @InProceedings{Dool98, author = "Dooley, Samuel S.", title = "Coordinating mathematical content and presentation markup in interactive mathematical documents", booktitle = "Proc. ISSAC 1998", series = "ISSAC 98", year = "1998", publisher = "ACM Press", location = "Rostock, Germany", pages = "13-15", keywords = "axiomref", abstract = "This paper presents a method for representing mathematical content and presentation markup in interactive mathematical documents that treats each view of the information on a separate and equal footing. By providing extensible, overridable, default mappings from content to presentation in a way that supports efficient mappings back from the presentation to the underlying content, a user interface for an interactive textbook has been implemented where the user interacts with high-quality presentation markup that supports user operations defined in terms of the mathematical content. In addition, the user interface can be insulated from content-specific information, while still being enabled to transfer that information to other programs for computation. This method has been employed to embed interactive mathematical content into the IBM techexplorer Interactive Textbook for Linear Algebra. The issues involved in the implementation of the interactive textbook also shed some light on the problems faced by the MathML working group in representing both presentation and content for mathematics for interactive web documents." } \end{chunk} \index{Dunstan, Martin} \index{Kelsey, Tom} \index{Linton, Steve A.} \index{Martin, Ursula} \begin{chunk}{axiom.bib} @InProceedings{Duns98, author = "Dunstan, Martin and Kelsey, Tom and Linton, Steve and Martin, Ursula", title = "Lightweight Formal Methods For Computer Algebra Systems", publisher = "ACM Press", booktitle = "Proc. ISSAC 1998", year = "1998", location = "Rostock, Germany", pages = "80-87", url = "http://www.cs.st-andrews.ac.uk/~tom/pub/issac98.pdf", paper = "Duns98.pdf", keywords = "axiomref", abstract = "Demonstrates the use of formal methods tools to provide a semantics for the type hierarchy of the Axiom computer algebra system, and a methodology for Aldor program analysis and verification. There are examples of abstract specifications of Axiom primitives." } \end{chunk} \index{Harrison, J.} \index{Thery, L.} \begin{chunk}{axiom.bib} @article{Harr98, author = "Harrison, J. and Thery, L.", title = "A Skeptic's approach to combining HOL and Maple", journal = "J. Autom. Reasoning", volume = "21", number = "3", pages = "279-294", year = "1998", keywords = "axiomref", paper = "Harr98.pdf", url = "http://www.cl.cam.ac.uk/~jrh13/papers/cas.ps.gz", abstract = "We contrast theorem provers and computer algebra systems, pointing out the advantages and disadvantages of each, and suggest a simple way to achieve a synthesis of some of the best features of both. Our method is based on the systematic separation of search for a solution and checking the solution, using a physical connection between systems. We describe the separation of proof search and checking in some detail, relating it to proof planning and to the complexity class NP, and discuss different ways of exploiting a physical link between systems. Finally, the method is illustrated by some concrete examples of computer algebra results proved formally in the HOL theorem prover with the aid of Maple." } \end{chunk} \index{Kerber, Manfred} \index{Kohlhase, Michael} \index{Volker, Sorge} \begin{chunk}{axiom.bib} @article{Kerb98, author = "Kerber, Manfred and Kohlhase, Michael and Volker, Sorge", title = "Integrating computer algebra into proof planning", journal = "J. Autom. Reasoning", volume = "21", number = "3", pages = "327-355", keywords = "axiomref", paper = "Kerb98.pdf", url = "http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.40.3914&rep=rep1&type=pdf", abstract = "Mechanized reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two different tasks proving and calculating. Even more important, proof and computation are often interwoven and not easily separable. In this article, we advocate an integration of computer algebra into mechanized reasoning systems at the proof plan level. This approach allows us to view the computer algebra algorithms as methods, that is, declarative representations of the problem-solving knowledge specific to a certain mathematical domain. Automation can be achieved in many cases by searching for a hierarchic proof plan at the method level by using suitable domain-specific control knowledge about the mathematical algorithms. In other words, the uniform framework of proof planning allows us to solve a large class of problems that are not automatically solvable by separate systems. Our approach also gives an answer to the correctness problems inherent in such an integration. We advocate an approach where the computer algebra system produces high-level protocol information that can be processed by an interface to derive proof plans. Such a proof plan in turn can be expanded to proofs at different levels of abstraction, so the approach is well suited for producing a high-level verbalized explication as well as for a low-level, machine-checkable, calculus-level proof. We present an implementation of our ideas and exemplify them using an automatically solved example." } \end{chunk} \index{Naudin, Patrice} \index{Quitte, Claude} \begin{chunk}{axiom.bib} @article{Naud98, author = "Naudin, Patrice and Quitte, Claude", title = "Univariate polynomial factorization over finite fields", journal = "Theor. Comput. Sci.", volume = "191", number = "1-2", pages = "1-36", year = "1998", paper = "Naud98.pdf", abstract = "This paper is a tutorial introduction to univariate polynomial factorization over finite fields. The authors recall the classical methods that induced most factorization algorithms (Berlekamp’s and the Cantor-Zassenhaus ones) and some refinements which can be applied to these methods. Explicit algorithms are presented in a form suitable for almost immediate implementation. They give a detailed description of an efficient implementation of the Cantor-Zassenhaus algorithm used in the release 2 of the Axiom computer algebra system." } \end{chunk} committed Jun 27, 2016 Commits on Jun 27, 2016 1. Goal: Axiom Literate Programming \index{Koepf, Wolfram} \begin{chunk}{axiom.bib} @InProceedings{Koep99, author = "Koepf, Wolfram", title = "Orthogonal polnomials and computer algebra", booktitle = "Recent developments in complex analysis and computer algebra", series = "ISSAC 97", year = "1999", publisher = "Kluwer Adademic Publishers", location = "Newark, DE", pages = "205-234", keywords = "axiomref", paper = "Koep99.pdf", abstract = "Orthogonal polynomials have a long history, and are still important objects of consideration in mathematical research as well as in applications in Mathematical Physics, Chemistry, and Engineering. Quite a lot is known about them. Particularly well-known are differential equations, recurrence equations, Rodrigues formulas, generating functions and hypergeometric representations for the classical systems of Jacobi, Laguerre and Hermite which can be found in mathematical dictionaries. Less well-known are the corresponding representations for the classical discrete systems of Hahn, Krawtchouk, Meixner and Charlier, as well as addition theorems, connection relations between different systems and other identities for these and other systems of orthogonal polynomials. The ongoing research in this still very active subject of mathematics expands the knowledge database about orthogonal polynomials continuously. In the last few decades the classical families have been extended to a rather large collection of polynomial systems, the so-called Askey-Wilson scheme, and they have been generalized in other ways as well. Recently new algorithmic approaches have been discovered to compute differential, recurrence and similar equations from series or integral representations. These methods turn out to be quite useful to prove or detect identities for orthogonal polynomial systems. Further algorithms to detect connection coefficients or to identify polynomial systems from given recurrence equations have been developed. Although some algorithmic methods had been known already in the last century, their use was rather limited due to the immense amount of calculations. Only the existence and distribution of computer algebra systems makes their use simple and useful for everybody. In this plenary lecture an overview is given of how algorithmic methods implemented in computer algebra systems can be used to prove identities about and to detect new knowledge for orthogonal polynomials and other hypergeometric type special functions. Implementations for this type of algorithms exist in Maple, Mathematica and REDUCE, and maybe also in other computer algebra systems. Online demonstrations will be given using Maple V.5." } \end{chunk} \index{Koepf, Wolfram} \begin{chunk}{axiom.bib} @misc{Koep14, author = "Koepf, Wolfram", title = "Methods of Computer Algebra for Orthogonal Polynomials", year = "2014", location = "Rutgers, NJ, USA", url = "http://www.mathematik.uni-kassel.de/~koepf/Vortrag/2014-Zeilberger-Vortrag.pdf", paper = "Koep14.pdf", video1 = "https://vimeo.com/85573338", video2 = "https://vimeo.com/85573712", website = "http://www.caop.org" } \end{chunk} \index{Daly, Timothy} \begin{chunk}{axiom.bib} @misc{Daly08, author = "Daly, Timothy", title = "Axiom Computer Algebra System Information Sources", video = "https://www.youtube.com/watch?v=CV8y3UrpadY", keywords = "axiomref", year = "2008" } \end{chunk} \index{Koepf, Wolfram} \begin{chunk}{axiom.bib} @article{Koep99a, author = "Koepf, Wolfram", title = "Software for the algorithmic work with orthogonal polynomials and special functions", journal = "Electron. Trans. Numer. Anal.", volume = "9", year = "1999", keywords = "axiomref", paper = "Koep99a.pdf", url = "http://arxiv.org/pdf/math/9809125v1.pdf", abstract = "An overview of the MAPLE routines that can be used for hypergeometric and basic hypergeometric series with some discussion of how and why they work." } \end{chunk} \index{Martin, Ursula} \begin{chunk}{axiom.bib} @InProceedings{Mart99, author = "Martin, Ursula", title = "Computers, reasoning and mathematical practice", booktitle = "Computational Logic", publisher = "Springer", year = "1999", location = "Marktoberdorf, Germany", pages = "301-346", keywords = "axiomref", paper = "Mart99.pdf", url = "http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.50.2061&rep=rep1&type=pdf", abstract = "We identify three main objectives for computer aided reasoning enhancing the techniques available for mathematical experimentation, developing community standards for experiment and modelling and developing methods which will make computation more acceptable as part of a proof. We discuss three areas of research which address these: the use of theorem proving techniques to enhance or extend mathematical software systems, support for formal methods techniques to increase the reliability of such systems, and the use of computer aided formal reasoning in support of mathematical practice. This last includes activities such as formalizing systems for computational mathematics or visualization so that they can still be used informally but generate a formal development, and developing techniques to provide assistance in the initial stages of developing a new theory. By mathematics here we mean the activities of working research mathematicians, producing new results in pure or applied mathematics, although we touch briefly on some questions concerning the applications of computational mathematics and simulation in research science and engineering. We have left out several other related areas entirely: logical questions of decidability, soundness or completeness, theoretical computer science issues of semantics, computability or complexity, foundational issues such as constructivity, computer aided proofs about software and hardware, and the use of computers in mathematical education at all levels and in heuristic discovery. In particular foundational questions about computation have transformed mathematical logic, computation has made constructive proof feasible, and effective notions of practice for proofs about hardware and software are by no means well understood. However these matters fall outside the scope of this paper." } \end{chunk} \index{Brown, Ronald} \index{Tonks, Andrew} \begin{chunk}{axiom.bib} @article{Brow94, author = "Brown, Ronald and Tonks, Andrew", title = "Calculations with simplicial and cubical groups in AXIOM", journal = "Journal of Symbolic Computation", volume = "17", number = "2", pages = "159-179", year = "1994", month = "February", misc = "CODEN JSYCEH ISSN 0747-7171", keywords = "axiomref", paper = "Brow94.pdf", abstract = "Work on calculations with simplical and cubical groups in AXIOM was carried out using loan equipment and software from IBM UK and guidance from L. A. Lambe. We report on the results of this work, and present the AXIOM code written by the second author during this period. This includes an implementation of the monoids which model cubes and simplices, together with a new AXIOM category of near-rings with which to carry out non-abelian calculations. Examples of the use of this code in interactive AXIOM sessions are also given." } \end{chunk} committed Jun 27, 2016 2. Goal: Axiom Literate Programming \index{Kumar, P.} \index{Pellegrino, S.} \begin{chunk}{axiom.bib} @article{Kuma00, author = "Kumar, P. and Pellegrino, S.", title = "Computation of kinematic paths and bifurcation points", journal = "Int. J. Solids Struct.", volume = "37", number = "46-47", pages = "7003-7027", year = "2000", keywords = "axiomref", abstract = "This article deals with the kinematic simulation of movable structures that go through special configurations of kinematic bifurcation, as they move. A series of algorithms are developed for structures that can be modelled using pin-jointed bars and that admit a single-parameter motion. These algorithms are able to detect and locate any bifurcation points that exist along the path of the structure and, at each bifurcation point, can determine all possible motions of the structure. The theory behind the algorithms is explained, and the analysis of a simple example is discussed in detail. Then, a simplified version of the particular problem that had motivated this work, the simulation of the folding and deployment of a thin membrane structure forming a solar sail, is analysed. For the particular cases that are considered, it is found that the entire process is inextensional, but a detailed study of the simulation results shows that in more general cases, it is likely that stretching or wrinkling will occur." } \end{chunk} \index{Safouhi, Hassan} \begin{chunk}{axiom.bib} @article{Safo00, author = "Safouhi, Hassan", title = {The$HD$and$H\overline{D}$methods for accelerating the convergence of three-center nuclear attraction and four-center two-electron Coulomb integrals over$B$functions and their convergence properties.}, journal = "J. Comput. Phys.", volume = "165", number = "2", pages = "473-495", year = "2000", keywords = "axiomref", abstract = "Three-center nuclear attraction and four-center two-electron Coulomb integrals over Slater-type orbitals are required for$ab$initio and density functional theory molecular structure calculations. They occur in many millions of terms, even for small molecules and require rapid and accurate evaluation. The$B$functions are used as a basis set of atomic orbitals. These functions are well adapted to the Fourier transform method that allowed analytical expressions for the integrals of interest to be developed. Rapid and accurate evaluation of these analytical expressions is now made possible by applying the$HD$and$H\overline{D}$methods for accelerating the convergence of the semi-infinite oscillatory integrals. The convergence properties of the new methods are analyzed. The numerical results section shows the high predetermined accuracy and the substantial gain in the calculation times obtained using the new methods." } \end{chunk} \index{Adams, A.A.} \index{Gottliebsen, H.} \index{Linton, S.A.} \index{Martin, U.} \begin{chunk}{axiom.bib} @InProceedings{Adam99a, author = "Adams, A.A. and Gottliebsen, H. and Linton, S.A. and Martin, U.", title = "VSDITLU: A verifiable symbolic definite integral table look-up", booktitle = "Automated Deduction", series = "CADE 16", year = "1999", location = "Trento, Italy", pages = "112-126", keywords = "axiomref", paper = "Adam99a.pdf", url = "http://www.a-cubed.info/Publications/CADE99.pdf", abstract = "We present a verifiable symbolic definite integral table lookup: 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 facts about the reals in the theorem prover PVS to aid in the transformation of the table entry into an answer. Our system is able to obtain correct answers in cases where standard techniques implemented in computer algebra systems fail. 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 do not. We extend upon Fateman’s web-based table by including parametric limits of integration and queries with side conditions." } \end{chunk} \index{Aitken, William E.} \index{Constable, Robert L.} \index{Underwood, Judith L.} \begin{chunk}{axiom.bib} @article{Aitk99, author = "Aitken, William E. and Constable, Robert L. and Underwood, Judith L.", title = "Metalogical frameworks. II: Developing a reflected decision procedure", journal = "J. Autom. Reasoning", volume = "22", number = "2", pages = "171-221", year = "1999", keywords = "axiomref", paper = "Aitk99.pdf", url = "http://www.nuprl.org/documents/Constable/MetalogicalFrameworksII.pdf", abstract = "Proving theorems is a creative act demanding new combinations of ideas and on occasion new methods of argument. For this reason, theorem proving systems need to be extensible. The provers should also remain correct under extension, so there must be a secure mechanism for doing this. The tactic-style provers pioneered by Edinburgh LCF provide a very effective way to achieve secure extensions, but in such systems, all new methods must be reduced to tactics. This is a drawback because there are other useful proof generating tools such as decision procedures; these include, for example, algorithms which reduce a deduction problem, such as arithmetic provability, to a computation on graphs. The Nuprl system pioneered the combination of fixed decision procedures with tactics, but the issue of securely adding new ones was not solved. In this paper, we show how to safely include user-defined decision procedures in theorem provers. The idea is to prove properties of the procedure inside the prover’s logic and then invoke a reflection rule to connect the procedure to the system. We also show that using a rich underlying logic permits an abstract account of the approach so that the results carry over to different implementations and other logics." } \end{chunk} \index{Boulm\'e, S.} \index{Hardin, T.} \index{Hirschkoff, D.} \index{Rioboo, Renaud} \index{M\'enissier-Morain, V.} \begin{chunk}{axiom.bib} @InProceedings{Boul99, author = "Boulme, S. and Hardin, T. and Hirschkoff, D. and Rioboo, Renaud", title = "On the way to certify Computer Algebra Systems", booktitle = "Systems for integrated computation and deduction", series = "Calculemus 99", year = "1999", publisher = "Elsevier", location = "Trento, Italy", pages = "11-12", keywords = "axiomref", paper = "Boul99.pdf", abstract = " The FOC project aims at supporting, within a coherent software system, the entire process of mathematical computation, starting with proved theories, ending with certified implementations of algorithms. In this paper, we explain our design requirements for the implementation, using polynomials as a running example. Indeed, proving correctness of implementations depends heavily on the way this design allows mathematical properties to be truly handled at the programming level. The FOC project, started at the fall of 1997, is aimed to build a programming environment for the development of certified symbolic computation. The working languages are Coq and Ocaml. In this paper, we present first the motivations of the project. We then explain why and how our concern for proving properties of programs has led us to certain implementation choices in Ocaml. This way, the sources express exactly the mathematical dependencies between different structures. This may ease the achievement of proofs." } \end{chunk} committed Jun 27, 2016 3. Goal: Correct Axiom Mathematics ========================================================================= todo 339: missing side conditions integrate((x-b)^(-1),x) (1) log(x - b) Type: Union(Expression(Integer),...) should show the side-condition x > b or should be log(abs(x-b)) committed Jun 27, 2016 4. Goal: Axiom Literate Programming \index{Prevosto, Virgile} \index{Doligez, Damien} \begin{chunk}{axiom.bib} @article{Prev02, author = "Prevosto, Virgile and Doligez, Damien", title = "Algorithms and proofs inheritance in the FOC language", journal = "J. Autom. Reasoning", volume = "29", number = "3-4", pages = "337-363", keywords = "axiomref", paper = "Prev02.pdf", abstract = "In this paper, we present the FOC language, dedicated to the development of certified computer algebra libraries (that is sets of programs). These libraries are based on a hierarchy of implementations of mathematical structures. After presenting the core set of features of our language, we describe the static analyses, which reject inconsistent programs. We then show how we translate FOC definitions into OCAML and COQ, our target languages for the computational part and the proof checking, respectively." } \end{chunk} \index{Schwarz, Fritz} \begin{chunk}{axiom.bib} @InProceedings{Schw02, author = "Schwarz, Fritz", title = "ALLTYPES: An algebraic language and type system", booktitle = "1st Int. Congress on Mathematical Software", year = "2002", isbn = "981-238-048-5", location = "Beijing China", pages = "486-500", keywords = "axiomref", paper = "Schw02.pdf", url = "http://www.scai.fraunhofer.de/content/dam/scai/de/documents/Mitarbeiterinnen-und-Mitarbeiter/ICMS2002.pdf", abstract = "The software system ALLTYPES provides an environment that is particularly designed for developing software in differential algebra. Its most important features may be described as follows: A set of about thirty parametrized algebraic types is defined. Data objects represented by these types may be manipulated by more than one hundred polymorphic functions. Reusability of code is achieved by genericity and multiple inheritance. The user may extend the system by defining new types and polymorphic functions. A language comprising seven basic language constructs is defined for implementing mathematical algorithms. The easy manipulation of types is particularly supported due to a special portion of the language dedicated to manipulating typed objects, i.e. for performing user-defined or automatic type coercions. Type inquiries are also included in the language." } \end{chunk} \index{Adams, Andrew A.} \index{Dunstan, Martin} \index{Gottlieben, Hanne} \index{Kelsey, Tom} \index{Martin, Ursula} \index{Owre, Sam} \begin{chunk}{axiom.bib} @InProceedings{Adam01, author = "Adams, Andrew A. and Dunstan, Martin and Gottlieben, Hanne and Kelsey, Tom and Martin, Ursula and Owre, Sam", title = "Computer algebra meets automated theorem proving: Integrating Maple and PVS", booktitle = "Theorem proving in higher order logics", series = "TPHOLs 2001", year = "2001", location = "Edinburgh, Scotland", pages = "27-42", keywords = "axiomref", paper = "Adam01.pdf", abstract = "We describe an interface between version 6 of the Maple computer algebra system with the PVS automated theorem prover. The interface is designed to allow Maple users access to the robust and checkable proof environment of PVS. We also extend this environment by the provision of a library of proof strategies for use in real analysis. We demonstrate examples using the interface and the real analysis library. These examples provide proofs which are both illustrative and applicable to genuine symbolic computation problems." } \end{chunk} \index{Antoine, Xavier} \begin{chunk}{axiom.bib} @article{Anto01, author = "Antoine, Xavier", title = "Microlocal diagonalization of strictly hyperbolic pseudodifferential systems and application to the design of radiation conditions in electromagnetism", journal = "SIAM J. Appl. Math", volume = "61", number = "6", pages = "1877-1905", year = "2001", keywords = "axiomref", abstract = "In [Commun. Pure Appl. Math. 28, 457-478 (1975; Zbl 0332.35058)], M. E. Taylor described a constructive diagonalization method to investigate the reflection of singularities of the solution to first-order hyperbolic systems. According to further works initiated by Engquist and Majda, this approach proved to be well adapted to the construction of artificial boundary conditions. However, in the case of systems governed by pseudodifferential operators with variable coefficients, Taylor’s method involves very elaborate calculations of the symbols of the operators. Hence, a direct approach may be difficult when dealing with high-order conditions. This motivates the first part of this paper, where a recursive explicit formulation of Taylor’s method is stated for microlocally strictly hyperbolic systems. Consequently, it provides an algorithm leading to symbolic calculations which can be handled by a computer algebra system. In the second part, an application of the method is investigated for the construction of local radiation boundary conditions on arbitrarily shaped surfaces for the transverse electric Maxwell system. It is proved that they are of complete order, provided the introduction of a new symbols class well adapted to the Maxwell system. Next, a complete second-order condition is designed, and when used as an on-surface radiation condition [G. A. Kriegsmann, A. Taflove, and K. R. Umashankar, IEEE Trans. Antennas Propag. 35, 153-161 (1987; Zbl 0947.78571)], it can give rise to an ultrafast numerical approximate solution of the scattered field." } \end{chunk} \index{Delliere, Stephane} \begin{chunk}{axiom.bib} @article{Dell01, author = "Delliere, Stephane", title = "On the links between triangular sets and dynamic constructable closure", journal = "J. Pure Appl. Algebra", volume = "163", number = "1", pages = "49-68", year = "2001", keywords = "axiomref", abstract = "Two kinds of triangular systems are studied: normalized triangular polynomial systems (a weaker form of Lazard’s triangular sets [D. Lazard, Discrete Appl. Math. 33, No. 1-3, 147-160 (1991; Zbl 0753.13013)] and constructible triangular systems (involved in the dynamic constructible closure programs of T. Gomez-Díaz [Quelques applications de l'evaluation dynamique, Ph.D. Thesis, Universite de Limoges (1994)]. This paper shows that these notions are strongly related. In particular, combining the two points of view (constructible and polynomial) on the subject of square-free conditions, it allows us to effect dramatic improvements in the dynamic constructible closure programs." } \end{chunk} \index{Michler, Gerhard O.} \begin{chunk}{axiom.bib} @article{Mich01, author = "Michler, Gerhard O.", title = "The character values of multiplicity-free irreducible constituents of a transitive permutation representation", journal = "Kyushu J. Math.", volume = "55", number = "1", pages = "75-106", year = "2001", keywords = "axiomref", url = "https://www.jstage.jst.go.jp/article/kyushujm/55/1/55\_1\_75/\_pdf", paper = "Mich01.pdf", abstract = "Let$G$be a finite group and$M$a subgroup of$G$. Let the permutation character of$G$on the set of right cosets of$M$be denoted by$(1_M)^G$. Any ordinary irreducible constituent of$(1_M)^G$with multiplicity one is called a multiplicity-free constituent of$(1_M)^G$. In the paper under review the author gives an efficient algorithm to compute the values of any multiplicity-free constituent of$(1_M)^G$. The character value is in terms of the double coset decomposition of$M$and related concepts concerning intersection matrices. The author uses this algorithm to determine the values of the multiplicity-free constituents of$(1_C)^{J_1}$, where$J_1$is the smallest Janko group of order 175560 and$C$is the centralizer of an involution in$J_1$. Using these, he then is able to compute the character table of$J_1$which is already known." } \end{chunk} \index{Rudnicki, Piotr} \index{Schwarzweller, Christoph} \index{Trybulec, Andrzej} \begin{chunk}{axiom.bib} @article{Rudn01, author = "Rudnicki, Piotr and Schwarzweller, Christoph and Trybulec, Andrzej", title = "Commutative algebra in the Mizar system", journal = "J. Symb. Comput.", volume = "32", number = "1-2", pages = "143-169", year = "2001", keywords = "axiomref", url = "https://inf.ug.edu.pl/~schwarzw/papers/jsc01.pdf", paper = "Rudn01.pdf", abstract = "We report on the development of algebra in the Mizar system. This includes the construction of formal multivariate power series and polynomials as well as the definition of ideals up to a proof of the Hilbert basis theorem. We present how the algebraic structures are handled and how we inherited the past developments from the Mizar Mathematical Library (MML). The MML evolves and past contributions are revised and generalized. Our work on formal power series caused a number of such revisions. It seems that revising past developments with an intent to generalize them is a necessity when building a database of formalized mathematics. This poses a question: how much generalization is best?" } \end{chunk} \index{Safouhi, Hassan} \begin{chunk}{axiom.bib} @article{Safo01, author = "Safouhi, Hassan", title = "Numerical evaluation of three-center two-electron Coulomb and hybrid integrals over$B$functions using the$HD$and$H\overline{D}$methods and convergence properties", journal = "J. Math. Chem.", volume = "29", number = "3", pages = "213-232", year = "2001", keywords = "axiomref", abstract = "The$B$function is a product of a spherical$K$Bessel function and a spherical harmonic. The integrals to be evaluated contain complicated expressions of finite hypergeometric functions and spherical$J$Bessel functions. Sequence transformation techniques are used for the numerical evaluation of the integrals. Numerical examples illustrate the accuracy and the efficiency of the method." } \end{chunk} \index{Thompson, Simon} \begin{chunk}{ignore} @InProceedings{Thom01, author = "Thompson, Simon", title = "Logic and dependent types in the Aldor Computer Algebra System", booktitle = "Symbolic computation and automated reasoning", series = "CALCULEMUS 2000", year = "2001", location = "St. Andrews, Scotland", pages = "205-233", keywords = "axiomref", paper = "Thom01.pdf", url = "http://axiom-wiki.newsynthesis.org/public/refs/aldor-calc2000.pdf", abstract = " We show how the Aldor type system can represent propositions of first-order 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} \index{Poll, Erik} \begin{chunk}{axiom.bib} @misc{Poll99, author = "Poll, Erik", title = "The Type System of Axiom", year = "1999", url = "http://www.cs.ru.nl/E.Poll/talks/axiom.pdf", paper = "Poll99.pdf", keywords = "axiomref", abstract = "This is a slide deck from a talk on the correspondence between Axiom/Aldor types and Logic." } \end{chunk} \index{Poll, Erik} \index{Thompson, Simon} \begin{chunk}{axiom.bib} @misc{Pollxx, author = "Poll, Erik and Thompson, Simon", title = "Adding the axioms to Axiom. Toward a system of automated reasoning in Aldor", url = "http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.1457&rep=rep1&type=ps", paper = "Pollxx.pdf", keywords = "axiomref", 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 post-conditions. The Curry-Howard 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} \index{Poll, Erik} \index{Thompson, Simon} \begin{chunk}{axiom.bib} @misc{Poll00, author = "Poll, Erik and Thompson, Simon", title = "Integrating Computer Algebra and Reasoning through the Type System of Aldor", paper = "P0ll00.pdf", keywords = "axiomref", year = "2000", 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 Curry-Howard isomorphism. We give a number of example applications of the logic we construct and explain a prototype implementation of a modified type-checking system written in Haskell." } \end{chunk} \index{Page, Bill} \begin{chunk}{axiom.bib} @misc{Page08, author = "Page, Bill", title = "Algebraist Network", url = "http://lambda-the-ultimate.org/node/2737", year = "2008", keywords = "axiomref" } \end{chunk} \index{Poll, Erik} \index{Thompson, Simon} \begin{chunk}{axiom.bib} @InProceedings{Poll00a, author = "Poll, Erik and Thompson, Simon", title = "Integrating Computer Algebra and Reasoning through the Type System of Aldor", booktitle = "Frontiers of Combining Systems", series = "Lecture Notes in Artificial Intelligence", year = "2000", isbn = "3-540-67281-8", location = "Nancy, France", pages = "136-150", keywords = "axiomref" } \end{chunk} \index{Davenport, James H.} \begin{chunk}{axiom.bib} @InProceedings{Dave00, author = "Davenport, James H.", title = "Abstract data types in computer algebra", booktitle = "Mathematical foundations of computer science", series = "MFCS 2000", year = "2000", location = "Bratislava, Slovakia", pages = "21-35", keywords = "axiomref", abstract = "The theory of abstract data types was developed in the late 1970s and the 1980s by several people, including the ADJ'' group, whose work influenced the design of Axiom. One practical manifestation of this theory was the OBJ-3 system. An area of computing that cries out for this approach is computer algebra, where the objects of discourse are mathematical, generally satisfying various algebraic rules. There have been various theoretical studies of this in the literature. The aim of this paper is to report on the practical applications of this theory within computer algebra, and also to outline some of the theoretical issues raised by this practical application. We also give a substantial bibliography." } \end{chunk} \index{Weber, Andreas} \begin{chunk}{axiom.bib} @phdthesis{Webe93b, author = "Weber, Andreas", title = "Type Systems for Computer Algebra", school = "University of Tubingen", year = "1993", paper = "Webe93b.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 order-sorted 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 first-order rewrite rules." } \end{chunk} \index{Fateman, Richard J.} \begin{chunk}{axiom.bib} @InProceedings{Fate00, author = "Fateman, Richard J.", title = "Problem solving environments and symbolic computing", booktitle = "Enabling technologies for computational science", publisher = "Kluwer Academic Publishers", year = "2000", pages = "91-102", keywords = "axiomref", paper = "Fate00.pdf", url = "http://people.eecs.berkeley.edu/~fateman/papers/pse-kluwer.pdf", abstract = "What role should be played by symbolic mathematical computation facilities in scientific and engineering problem solving environments''? Drawing upon standard facilities such as numerical and graphical libraries, symbolic computation should be useful for: The creation and manipulation of mathematical models; The production of custom optimized numerical software; The solution of delicate classes of mathematical problems that require handling beyond that available in traditional machine-supported floating-point computation. Symbolic representation and manipulation can potentially play a central organizing role in PSEs since their more general object representation allows a program to deal with a wider range of computational issues. In particular numerical, graphical, and other processing can be viewed as special cases of symbolic manipulation with interactive symbolic computing providing both an organizing backbone and the communication glue'' among otherwise dissimilar components" } \end{chunk} \index{Hoang, Ngoc Minh} \index{Petitot, Michel} \index{Van er Hoeven, Joris} \begin{chunk}{axiom.bib} @article{Hoan00, author = "Hoang, Ngoc Minh and Petitot, Michel and Van er Hoeven, Joris", title = "Shuffle algebra and polylogarithms", journal = "Discrete Math.", volume = "225", number = "1-3", pages = "217-230", year = "2000", keywords = "axiomref", paper = "Hoan00.pdf", abstract = "Generalized polylogarithms are defined as iterated integrals with respect to the two differential forms$\omega_0=\frac{dz}{z}$and$\omega_1=\frac{dz}{(1-z)}$. We give an algorithm which computes the monodromy of these special functions. This algorithm, implemented in AXIOM, is based on the computation of the associator$\Phi_{KZ}$of Drinfel’d, in factorized form. The monodromy formulae involve special constants, called multiple zeta values. We prove that the algebra of polylogarithms is isomorphic to a shuffle algebra." } \end{chunk} \index{Raab, C.G.} \begin{chunk}{axiom.bib} @article{Raab13, author = "Raab, C.G.", title = "Generalization of Risch's Algorithm to Special Functions", journal = "CoRR", volume = "abs/1305.1481", year = "2013", url = "http://arxiv.org/pdf/1305.1481v1.pdf", paper = "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 ideas 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} \index{Houstis, Elias N.} \index{Rice, John R.} \begin{chunk}{axiom.bib} @article{Hous00, author = "Houstis, Elias N. and Rice, John R.", title = "Future problem solving environments for computational science", journal = "Math. Comput. Simul.", volume = "54", number = "4-5", pages = "243-257", year = "2000", keywords = "axiomref", abstract = "We review the current state of the Problem Solving Environment (PSE) field and make projections for the future. First, we describe the computing context, the definition of a PSE and the goals of a PSE. The state-of-the-art is summarized along with the principal components and paradigms for building PSEs. The discussion of the future is given in three parts: future trends, scenarios for 2010/2025, and research issues to be addressed." } \end{chunk} \index{Gallopoulos, Stratis} \index{Houstis, Elias} \index{Rice, John} \begin{chunk}{axiom.bib} @techreport{Gall92, author = "Gallopoulos, Stratis and Houstis, Elias and Rice, John", title = "Future Research Directions in Problem Solving Environments for Computational Science", institution = "Purdue University", year = "1992", type = "technical report", number = "CSD-TR-92-032", paper = "Gall92.pdf", url = "http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1953\&context=cstech", keywords = "axiomref", abstract = "During the early 19605 some were visualizing that computers could provide a powerful problem solving environment (PSE) which would interact with scientists on their own terms. By the mid 1960s there were many attempts underway to create these PSEs, but the early 1970s almost all of these attempts had been abandoned, because the technological infrastructure could not yet support PSEs in computational science. The dream of the 1960s can be the reality of the 1990s; high performance computers combined with better understanding of computing and computational science have put PSEs well within our reach." } \end{chunk} committed Jun 27, 2016 Commits on Jun 26, 2016 1. Goal: Axiom Literate Programming \index{Tournier, Evelyne} \begin{chunk}{axiom.bib} @misc{Tour95, author = "Tournier, Evelyne", title = "Summary of organisation, history and possible future", paper = "Tour95.pdf", year = "1995", url = "ftp://ftp.inf.ethz.ch/org/cathode/workshops/jan95/abstracts/tournier.ps", keywords = "axiomref" } \end{chunk} \index{Watt, Stephen M.} \begin{chunk}{axiom.bib} @misc{Watt95a, author = "Watt, Stephen M.", title = "The A\# Programming Language and Compiler", paper = "Watt95a.pdf", year = "1995", url = "ftp://ftp.inf.ethz.ch/org/cathode/workshops/march93/Watt.tex", keywords = "axiomref" } \end{chunk} \index{Lecerf, Gr{\'e}goire} \begin{chunk}{axiom.bib} @article{Lece02, author = "Lecerf, Gregoire", title = "Quadratic Newton iteration for systems with multiplicity", journal = "Found. Comput. Math.", volume = "2", number = "3", pages = "247-293", year = "2002", keywords = "axiomref", paper = "Lece02.pdf", abstract = "The author proposes an efficient iterator with quadratic convergence that generalizes Newton iterator for multiple roots. It is based on an$m$-adic topology where the ideal$m$can be chosen generic enough. Compared to the Newton iterator the proposed iterator introduces a small overhead that grows with the square of the multiplicity of the root." } \end{chunk} \index{Lecerf, Gr{\'e}goire} \begin{chunk}{axiom.bib} @article{Lece96, author = "Lecerf, Gregoire", title = "Dynamic Evaluation and Real Closure Implementation in Axiom", year = "1996", url = "http://lecerf.perso.math.cnrs.fr/software/drc/drc.ps", paper = "Lece96.pdf", keywords = "axiomref" } \end{chunk} \index{Lomonaco, Samual J.} \index{Kauffman, Louis H.} \begin{chunk}{axiom.bib} @InProceedings{Lomo02, author = "Lomonaco, Samual J. and Kauffman, Louis H.", title = "Quantum hidden subgroup algorithms: a mathematical perspective", booktitle = "Quantum computation and information", series = "AMS special session", year = "2002", isbn = "0-8218-2140-7", location = "Washington", pages = "139-202", keywords = "axiomref", paper = "Lomo02.pdf", url = "https://arxiv.org/pdf/quant-ph/0201095v3.pdf", abstract = "The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of abelian algorithms of the Shor/Simon genre. Our strategy is to make this class of algorithms as mathematically transparent as possible. By the phrase mathematically transparent'' we mean to expose, to bring to the surface, and to make explicit the concealed mathematical structures that are inherently and fundamentally a part of such algorithms. In so doing, we create symbolic abelian quantum hidden subgroup algorithms that are analogous to the those symbolic algorithms found within such software packages as Axiom, Cayley, Maple, Mathematica, and Magma. As a spin-off of this effort, we create three different generalizations of Shor’s quantum factoring algorithm to free abelian groups of finite rank. We refer to these algorithms as wandering (or vintage$\mathbb{Z}_Q$) Shor algorithms. They are essentially quantum algorithms on free abelian groups of finite rank$n$which, with each iteration, first select a random cyclic direct summand$\mathbb{Z}$of the group and then apply one iteration of the standard Shor algorithm to produce a random character of the “approximating” finite group$\widetilde{A} = \mathbb{Z}_Q$, called the group probe. These characters are then in turn used to find either the order$P$of a maximal cyclic subgroup$\mathbb{Z}_P$of the hidden quotient group$H_{\varphi}$, or the entire hidden quotient group$H_{\varphi}$. An integral part of these wandering quantum algorithms is the selection of a very special random transversal$t_{\mu}:\widetilde{A}\rightarrow A$, which we refer to as a Shor transversal. The algorithmic time complexity of the first of these wandering Shor algorithms is found to be$O(n^2({rm lg\ } Q)^3 ({\rm lg\ }{\rm lg\ } Q)^{n+1})$." } \end{chunk} committed Jun 26, 2016 2. Goal: Axiom Literate Programming \index{Mulders, Thom} \begin{chunk}{axiom.bib} @misc{Muld95, author = "Mulders, Thom", title = "Primitives: Orepoly and Lodo", paper = "Muld95.pdf", year = "1995", url = "ftp://ftp.inf.ethz.ch/org/cathode/workshops/jan95/abstracts/mulders.ps", keywords = "axiomref", comment = "\newline\refto{category OREPCAT UnivariateSkewPolynomialCategory} \newline\refto{category LODOCAT LinearOrdinaryDifferentialOperatorCategory} \newline\refto{domain AUTOMOR Automorphism} \newline\refto{domain ORESUP SparseUnivariateSkewPolynomial} \newline\refto{domain OREUP UnivariateSkewPolynomial} \newline\refto{domain LODO LinearOrdinaryDifferentialOperator} \newline\refto{domain LODO1 LinearOrdinaryDifferentialOperator1} \newline\refto{domain LODO2 LinearOrdinaryDifferentialOperator2} \newline\refto{package APPLYORE ApplyUnivariateSkewPolynomial} \newline\refto{package OREPCTO UnivariateSkewPolynomialCategoryOps} \newline\refto{package LODOF LinearOrdinaryDifferentialOperatorFactorizer} \newline\refto{package LODOOPS LinearOrdinaryDifferentialOperatorsOps}" } \end{chunk} committed Jun 26, 2016 3. Goal: Axiom Literate Programming \index{Dewar, Mike C.} \begin{chunk}{axiom.bib} @misc{Dewa95, author = "Dewar, Mike C.", title = "AXIOM and A\#: Current Status and Future Plans", paper = "Dewa95.pdf", year = "1995", keywords = "axiomref", url = "ftp://ftp.inf.ethz.ch/org/cathode/workshops/jan95/abstracts/dewar.ps" } \end{chunk} \index{Dicrescenzo, C.} \index{Jung, Francoise} \begin{chunk}{axiom.bib} @misc{Dicr95, author = "Dicrescenzo, C. and Jung, Francoise", title = "COMPASS package", paper = "Dicr95.pdf", year = "1995", url = "ftp://ftp.inf.ethz.ch/org/cathode/workshops/jan95/abstracts/bronstein.ps", keywords = "axiomref" } \end{chunk} committed Jun 26, 2016 4. Goal: Axiom Literate Programming \index{Abramov, Sergei A.} \index{Bronstein, Manuel} \begin{chunk}{axiom.bib} @article{Abra01, author = "Abramov, Sergei and Bronstein, Manuel", title = "On Solutions of Linear Functional Systems", url = "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html", paper = "Abra01.pdf", comment = "\newline\refto{category OREPCAT UnivariateSkewPolynomialCategory} \newline\refto{category LODOCAT LinearOrdinaryDifferentialOperatorCategory} \newline\refto{domain AUTOMOR Automorphism} \newline\refto{domain ORESUP SparseUnivariateSkewPolynomial} \newline\refto{domain OREUP UnivariateSkewPolynomial} \newline\refto{domain LODO LinearOrdinaryDifferentialOperator} \newline\refto{domain LODO1 LinearOrdinaryDifferentialOperator1} \newline\refto{domain LODO2 LinearOrdinaryDifferentialOperator2} \newline\refto{package APPLYORE ApplyUnivariateSkewPolynomial} \newline\refto{package OREPCTO UnivariateSkewPolynomialCategoryOps} \newline\refto{package LODOF LinearOrdinaryDifferentialOperatorFactorizer} \newline\refto{package LODOOPS LinearOrdinaryDifferentialOperatorsOps}", abstract = " We describe a new direct algorithm for transforming a linear system of recurrences into an equivalent one with nonsingular leading or trailing matrix. Our algorithm, which is an improvement to the EG elimination method, uses only elementary linear algebra operations (ranks, kernels, and determinants) to produce an equation satisfied by the degress of the solutions with finite support. As a consequence, we can boudn and compute the polynomial and rational solutions of very general linear functional systems such as systems of differential or ($q$)-difference equations." } \end{chunk} committed Jun 26, 2016 5. Goal: Axiom Literate Programming \index{Bronstein, Manuel} \begin{chunk}{axiom.bib} @misc{Bron95, author = "Bronstein, Manuel", title = "On radical solutions of linear ordinary differential equations", paper = "Bron95.pdf", year = "1995", url = "ftp://ftp.inf.ethz.ch/org/cathode/workshops/jan95/abstracts/bronstein.ps", comment = "\newline\refto{category OREPCAT UnivariateSkewPolynomialCategory} \newline\refto{category LODOCAT LinearOrdinaryDifferentialOperatorCategory} \newline\refto{domain AUTOMOR Automorphism} \newline\refto{domain ORESUP SparseUnivariateSkewPolynomial} \newline\refto{domain OREUP UnivariateSkewPolynomial} \newline\refto{domain LODO LinearOrdinaryDifferentialOperator} \newline\refto{domain LODO1 LinearOrdinaryDifferentialOperator1} \newline\refto{domain LODO2 LinearOrdinaryDifferentialOperator2} \newline\refto{package APPLYORE ApplyUnivariateSkewPolynomial} \newline\refto{package OREPCTO UnivariateSkewPolynomialCategoryOps} \newline\refto{package LODOF LinearOrdinaryDifferentialOperatorFactorizer} \newline\refto{package LODOOPS LinearOrdinaryDifferentialOperatorsOps}" } \end{chunk} committed Jun 26, 2016 6. Goal: Axiom Literate Programming \index{Jacquemard, A.} \index{Teixeira, M.A.} \begin{chunk}{axiom.bib} @article{Jacq02, author = "Jacquemard, A. and Teixeira, M.A.", title = "Effective algebraic geometry and normal forms of reversible mappings", journal = "Rev. Mat. Complut.", volume = "15", number = "1", pages = "31-55", year = "2002", keywords = "axiomref", abstract = "The authors consider a problem coming from the theory of discrete dynamical systems (iterations of diffeomorphisms in$\mathbb{R}^3$). Namely, how to characterize the behaviour of the trajectories near the fixed points and the stability of this behaviour under perturbations. The authors use the computer in the context of reversible mappings. When such a mapping has some degree of degeneracy they apply Gröbner bases techniques and the theory of (formal) normal forms in the space of the coefficients of the jets of the mapping. They show that a language with typed objects like AXIOM is very convenient to solve such problems." } \end{chunk} \index{Lambe, Larry A.} \index{Seiler, Werner M.} \begin{chunk}{axiom.bib} @article{Lamb02, author = "Lambe, Larry A. and Seiler, Werner M.", title = "Differential equations, Spencer cohomology, and computing resolutions", journal = "Georgian Mathematical Journal", volume = "9", number = "4", pages = "723-774", year = "2002", keywords = "axiomref", paper = "Lamb02.pdf", url = "https://www.emis.de/journals/GMJ/vol9/v9n4-11.pdf", abstract = "Spencer cohomology is one of the classical tools in the study of overdetermined systems of partial differential equations. The paper proposes a new point of view of the Spencer cohomology based on a dual approach via comodules that allows to relate the Spencer cohomology to standard constructions in homological algebra, and discuss concrete methods for its construction based on homological perturbation theory. It also gives a detailed introduction to the subject, and proposes a new algorithm of constructing the Spencer resolution by using symbolic computation systems." } \end{chunk} \index{Davenport, James H.} \begin{chunk}{axiom.bib} @misc{Dave93, author = "Davenport, James H.", title = "The PoSSo Project", paper = "Dave93.pdf", keywords = "axiomref" } \end{chunk} \index{Bronstein, Manuel} \begin{chunk}{axiom.bib} @misc{Bron95, author = "Bronstein, Manuel", title = "On radical solutions of linear ordinary differential equations", paper = "Bron95.pdf", year = "1995", url = "ftp://ftp.inf.ethz.ch/org/cathode/workshops/jan95/abstracts/bronstein.ps" } \end{chunk} committed Jun 26, 2016 7. Goal: Axiom Literate Programming \index{Cohen, Joel S.} \begin{chunk}{axiom.bib} @book{Cohe03a, author = "Cohen, Joel S.", title = "Computer algebra and symbolic computation. Mathematical Methods", year = "2003", publisher = "A. K. Peters", isbn = "1-56881-159-4", keywords = "axiomref" } \end{chunk} \index{Cohen, Joel S.} \begin{chunk}{axiom.bib} @book{Cohe03b, author = "Cohen, Joel S.", title = "Computer algebra and symbolic computation. Elementary Algorithms", year = "2003", publisher = "A. K. Peters", isbn = "1-56881-159-4", keywords = "axiomref" } \end{chunk} \index{Farmer, William M.} \index{von Mohrenschildt, Martin} \begin{chunk}{axiom.bib} @article{Farm03, author = "Farmer, William M. and von Mohrenschildt, Martin", title = "An overview of a formal framework for managing mathematics", journal = "Ann. Math. Artif. Intell.", volume = "38", number = "1-3", pages = "165-191", year = "2003", keywords = "axiomref", paper = "Farm03.pdf", url = "https://www.emis.de/proceedings/MKM2001/farmer.ps", abstract = "Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform theory which is simultaneously an axiomatic theory and an algorithmic theory. Representing a collection of mathematical models, a biform theory provides a formal context for both deduction and computation. The framework includes facilities for deriving theorems via a mixture of deduction and computation, constructing sound deduction and computation rules, and developing networks of biform theories linked by interpretations. The framework is not tied to a specific underlying logic; indeed, it is intended to be used with several background logics simultaneously. Many of the ideas and mechanisms used in the framework are inspired by the IMPS Interactive Mathematical Proof System and the Axiom computer algebra system." } \end{chunk} \index{Lamban, Laureano} \index{Pascual, Vico} \index{Rubio, Julio} \begin{chunk}{axiom.bib} @article{Lamb03, author = "Lamban, Laureano and Pascual, Vico and Rubio, Julio", title = "An object-oriented interpretation of the EAT system", journal = "Appl. Algebra Eng. Commun. Comput.", volume = "14", number = "3", pages = "187-215", keywords = "axiomref", abstract = "In a previous paper we characterized, in the category theory setting, a class of implementations of abstract data types, which has been suggested by the way of programming in the EAT system. (EAT, Effective Algebraic Topology, is one of Sergeraert’s systems for effective homology and homotopy computation.) This characterization was established using classical tools, in an unrelated way to the current mainstream topics in the field of algebraic specifications. Looking for a connection with these topics, we have found, rather unexpectedly, that our approach is related to some object-oriented formalisms, namely hidden specifications and the coalgebraic view. In this paper, we explore these relations making explicit the implicit object-oriented features of the EAT system and generalizing the data structure analysis we had previously done." } \end{chunk} \index{Barnett, Michael P.} \begin{chunk}{axiom.bib} @article{Barn02, author = "Barnett, Michael P.", title = "Computer algebra in the life sciences", journal = "SIGSAM Bulletin", volume = "36", number = "4", pages = "5-31", year = "2002", keywords = "axiomref", paper = "Barn02.pdf", url = "https://notendur.hi.is/vae11/\%C3\%9Eekking/Systems\%20Biology/Biological\%20Algebra.PDF", abstract = "This note (1) provides references to recent work that applies computer algebra (CA) to the life sciences, (2) cites literature that explains the biological background of each application, (3) states the mathematical methods that are used, (4) mentions the benefits of CA, and (5) suggests some topics for future work." } \end{chunk} \index{Roanes-Lozano, Eugenio} \index{Roanes-Macias, Eugenio} \index{Villar-Mena, M.} \begin{chunk}{axiom.bib} @article{Roan03, author = "Roanes-Lozano, Eugenio and Roanes-Macias, Eugenio and Villar-Mena, M.", title = "A bridge between dynamic geometry and computer algebra", journal = "Math. Comput. Modelling", volume = "37", number = "9-10", pages = "1005-1028", year = "2003", keywords = "axiomref", url = "ac.els-cdn.com/S0895717703001158/1-s2.0-S0895717703001158-main.pdf", paper = "Roan03.pdf", abstract = "Both Computer Algebra Systems (CASs) and dynamic geometry systems (DGSs) have reached a high level of development. Some CASs (like Maple or Derive) include specific and powerful packages devoted to Euclidean geometry, but CASs have incorporated neither mouse drawing capabilities nor dynamic capabilities. Meanwhile, the well-known DGSs do not provide algebraic facilities. Maple’s and Derive’s paramGeo packages and the DGS-CAS translator (all freely available from the authors) make it possible to draw a geometric configuration with the mouse (using The Geometer’s Sketchpad 3 or 4) and to obtain the coordinates, equations, etc., of the drawn configuration in Maple’s or Derive’s syntax. To obtain complicated formulae, coordinates of points or equations of loci, to perform automatic theorem proving and to perform automatic discovery directly from sketches are examples of straightforward applications. Moreover, this strategy could be adapted to other CASs and DGSs. This work clearly has a didactic application in geometric problems exploration. Nevertheless, its main interest is to provide a convenient time-saving way to introduce data when dealing with rule and compass geometry, which has a wider scope than only educational purposes." } \end{chunk} \index{Davenport, James H.} \begin{chunk}{axiom.bib} @article{Dave02, author = "Davenport, James H.", title = "Equality in computer algebra and beyond", journal = "J. Symbolic Computing", volume = "34", number = "4", pages = "259-270", year = "2002", keywords = "axiomref", paper = "Dave02.pdf", url = "http://www.calculemus.net/meetings/siena01/Papers/Davenport.pdf", abstract = "Equality is such a fundamental concept in mathematics that, in fact, we seldom explore it in detail, and tend to regard it as trivial. When it is shown to be non-trivial, we are often surprised. As is often the case, the computerization of mathematical computation in computer algebra systems on the one hand, and mathematical reasoning in theorem provers on the other hand, forces us to explore the issue of equality in greater detail.In practice, there are also several ambiguities in the definition of equality. For example, we refer to$\mathbb{Q}(x)$as rational functions'', even though$\frac{x^2-1}{x-1}$and$x+1$are not equal as functions from$\mathbb{R}$to$\mathbb{R}$, since the former is not defined at$x=1$, even though they are equal as elements of$\mathbb{Q}(x)$. The aim of this paper is to point out some of the problems, both with mathematical equality and with data structure equality, and to explain how necessary it is to keep a clear distintion between the two." } \end{chunk} committed Jun 26, 2016 8. Goal: Axiom Literate Programming committed Jun 26, 2016 9. Goal: Axiom Literate Programming \index{Gottlieben, Hanne} \index{Kelsey, Tom} \index{Martin, Ursula} \begin{chunk}{axiom.bib} @article{Gott05, author = "Gottliebsen, Hanne and Kelsey, Tom and Martin, Ursula", title = "Hidden verification for computational mathematics", journal = "Journal of Symbolic Computation", volume = "39", number = "5", pages = "539-567", year = "2005", url = "http://www.sciencedirect.com/science/article/pii/S0747717105000295", paper = "Gott05.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 pipe-lined 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 look-up tables." } \end{chunk} \index{Kaplan, Michael} \begin{chunk}{axiom.bib} @book{Kapl05, author = "Kaplan, Michael", title = "Computer Algebra", publisher = "Springer, Berlin, Germany", year = "2005", isbn = "3-540-21379-1", keywords = "axiomref", comment = "German Language" } \end{chunk} \index{Oancea, Cosmin E.} \index{Watt, Stephen M.} \begin{chunk}{axiom.bib} @InProceedings{Oanc05, author = "Oancea, Cosmin E. and Watt, Stephen M.", title = "Domains and expressions: an interface between two approaches to computer algebra", booktitle = "Proc. 2005 ISSAC", series = "ISSAC 2005", year = "2005", isbn = "1-59593-095-7", location = "Beijing, China", pages = "261-268", keywords = "axiomref", paper = "Oanc05.pdf", url = "http://www.csd.uwo.ca/~watt/pub/reprints/2005-issac-alma.pdf", abstract = "This paper describes a method to use compiled, strongly typed Aldor domains in the interpreted, expression-oriented Maple environment. This represents a non-traditional approach to structuring computer algebra software: using an efficient, compiled language, designed for writing large complex mathematical libraries, together with a top-level system based on user-interface priorities and ease of scripting. We examine what is required to use Aldor libraries to extend Maple in an effective and natural way. Since the computational models of Maple and Aldor differ significantly, new run-time code must implement a non-trivial semantic correspondence. Our solution allows Aldor functions to run tightly coupled to the Maple environment, able to directly and efficiently manipulate Maple data objects. We call the overall system Alma." } \end{chunk} \index{Brunelli, J.C.} \begin{chunk}{axiom.bib} \article{Brun04, author = "Brunelli, J.C.", title = "PSEUDO: applications of streams and lazy evaluation to integrable models", journal = "Comput. Phys. Commun.", volume = "163", number = "1", pages = "22-40", year = "2004", keywords = "axiomref", url = paper = abstract = "Procedures to manipulate pseudo-differential operators in MAPLE are implemented in the program PSEUDO to perform calculations with integrable models. We use lazy evaluation and streams to represent and operate with pseudo-differential operators. No order of truncation is needed since terms are produced on demand. We give a series of concrete examples." } \end{chunk} \index{Emiris, Ioannis Z.} \index{Tsigaridas, Elias P.} \begin{chunk}{axiom.bib} @InProceedings{Emir04, author = "Emiris, Ioannis Z. and Tsigaridas, Elias P.", title = "Comparing real algebraic numbers of small degree", booktitle = "12th annual European symposium", series = "ESA 2004", year = "2004", isbn = "3-540-23025-4", location = "Bergen, Norway", pages = "652-663", keywords = "axiomref", paper = "Emir04.pdf", url = "http://www-polsys.lip6.fr/~elias/files//et-esa-04.pdf", comment = "\newline\refto{domain RECLOS RealClosure}", abstract = "We study polynomials of degree up to 4 over the rationals or a computable real subfield. Our motivation comes from the need to evaluate predicates in nonlinear computational geometry efficiently and exactly. We show a new method to compare real algebraic numbers by precomputing generalized Sturm sequences, thus avoiding iterative methods; the method, moreover handles all degenerate cases. Our first contribution is the determination of rational isolating points, as functions of the coefficients, between any pair of real roots. Our second contribution is to exploit invariants and Bezoutian subexpressions in writing the sequences, in order to reduce bit complexity. The degree of the tested quantities in the input coefficients is optimal for degree up to 3, and for degree 4 in certain cases. Our methods readily apply to real solving of pairs of quadratic equations, and to sign determination of polynomials over algebraic numbers of degree up to 4. Our third contribution is an implementation in a new module of library SYNAPS v2.1. It improves significantly upon the efficiency of certain publicly available implementations: Rioboo’s approach on AXIOM, the package of Guibas-Karavelas-Russel, and CORE v1.6, MAPLE v9, and SYNAPS v2.0. Some existing limited tests had shown that it is faster than commercial library LEDA v4.5 for quadratic algebraic numbers." } \end{chunk} \index{Rioboo, Renaud} \begin{chunk}{axiom.bib} @InProceedings{Riob92, author = "Rioboo, Renaud", title = "Real algebraic closure of an ordered field, Implementation in Axiom", booktitle = "Proc. ISSAC 92", series = "ISSAC 92", year = "1992", isbn = "978-3-540-75186-1", location = "Berkeley, California", pages = "206-215", isbn = "0-89791-489-9 (soft), 0-89791-490-2 (hard)", paper = "Riob92.pdf", keywords = "axiomref", comment = "\newline\refto{domain RECLOS RealClosure}", 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} \index{Hivert, Florent} \index{Thiery, Nicolas M.} \begin{chunk}{axiom.bib} @article{Hive04, author = "Hivert, Florent and Thiery, Nicolas M.", title = "MuPAD-Combinat, an open-source package for research in algebraic combinatorics", journal = "Seminaire Lotharingien de Combinatoire", volume = "51", number = "B51z", year = "2004", keywords = "axiomref", url = "http://www.emis.de/journals/SLC/wpapers/s51thiery.pdf", paper = "Hive04.pdf", abstract = "In this article we give an overview of the MuPAD-Combinat open-source algebraic combinatorics package for the computer algebra system MuPAD 2.0.0 and higher. This includes our motivations for developing yet another combinatorial software, a tutorial introduction with lots of examples, as well as notes on the general design. The material presented here is also available as a part of the MuPAD-Combinat handbook; further details and references on the algorithms used can be found there. The package and the handbook are available from the web page, together with download and installation instructions, mailing lists, etc." } \end{chunk} \index{Hemmecke, Ralf} \index{Rubey, Martin} \begin{chunk}{axiom.bib} @misc{RISC06, title = "AXIOM Workshop 2006", url = "http://axiom-wiki.newsynthesis.org/WorkShopRISC2006", year = "2006", location = "Hagenberg, Austria", keywords = "axiomref", abstract = "Axiom is a computer algebra system with a long tradition. It recently became free software. The workshop aims at a cooperation of Axiom developers with developers of packages written for other Computer Algebra Systems or developers of stand-alone packages. Furthermore, the workshop wants to make the potential of Axiom and Aldor more widely known in order to attract new users and new developers." } \end{chunk} \index{Hemmecke, Ralf} \index{Rubey, Martin} \begin{chunk}{axiom.bib} @misc{RISC07, title = "AXIOM Workshop 2007", url = "http://axiom-wiki.newsynthesis.org/WorkShopRISC2007", year = "2007", location = "Hagenberg, Austria", keywords = "axiomref", abstract = "The workshop aims at a cooperation of Axiom developers with deelopers of packages written for other Computer Algebra Systems, and mathematicians that would like to use a computer algebra system to perform experiments. One goal of the workshop is to learn about the mathematical theory, the design of packages written for other CAS and to make those functionalities available in Axiom." } \end{chunk} committed Jun 26, 2016 10. Goal: Axiom Literate Programming committed Jun 25, 2016 11. Goal: Axiom Literate Programming \index{Carette, Jacques} \begin{chunk}{axiom.bib} @article{Care06, author = "Carette, Jacques", title = "Gaussian elimination: a case study in efficient genericity with MetaOCaml", journal = "Sci. Comput. Program.", volume = "62", number = "1", pages = "3-24", year = "2006", keywords = "axiomref", url = "http://www.cas.mcmaster.ca/~carette/publications/ge.pdf", paper = "Care06.pdf", abstract = "The Gaussian Elimination algorithm is in fact an algorithm family – common implementations contain at least six (mostly independent) design choices''. A generic implementation can easily be parametrized by all these design choices, but this usually leads to slow and bloated code. Using MetaOCaml’s staging facilities, we show how we can produce a natural and type-safe implementation of Gaussian Elimination which exposes its design choices at code-generation time, so that these choices can effectively be specialized away, and where the resulting code is quite efficient." } \end{chunk} \index{Li, Xin} \begin{chunk}{axiom.bib} @phdthesis{Lixx05, author = "Li, Xin", title = "Efficient Management of Symbolic Computation with Polynomials", school = "University of Western Ontario", year = "2005", paper = "Lixx05.pdf", keywords = "axiomref", url = "http://www.csd.uwo.ca/~moreno//Publications/XinLi-MSThesis-2005.pdf.gz", abstract = "Symbolic polynomial computation is widely used to solve many applied or abstract mathematical problems. Some of them, such as solving systems of polynomial equations, have exponential complexity. Their implementation is, therefore, a challenging task. By using adapted data structures, asymptotically fast algorithms and effective code optimization techniques, we show how to reduce the practical and theoretical complexity of these computations. Our effort is divided into three categories: integrting the best known techniques into our implementation, investigating new directions, and measuring the interactions between numerous techniques. We chose AXIOM and ALDOR as our implementation and experimentation environment, since they are both strongly typed and highly efficient Computer Algebra Systems (CAS). Our implementation results show that our methods have great potential to improve the efficiency of exact polynomial computations with the selected CASs. The performance of our implementation is comparable to that of (often outperforming) the best available packages for polynomial computations." } \end{chunk} \index{Li, Xin} \begin{chunk}{axiom.bib} @phdthesis{Lixx09a, author = "Li, Xin", title = "Toward High-Performance Polynomial System Solvers Based On Triangluar Decompositions", school = "University of Western Ontario", year = "2009", paper = "Lixx09a.pdf", keywords = "axiomref", url = "http://www.csd.uwo.ca/~moreno/Publications/XinLiPhDThesis-2008.pdf", abstract = "This thesis is devoted to the design and implementation of polynomial system solvers based on symbolic computation. Solving systems of non-linear, algebraic or differential equations, is a fundamental problem in mathematical sciences. It has been studied for centuries and still stimulates many research developments, in particular on the front of high-performance computing. Triangular decompositions are a highly promising technique with the potential to produce high-performance polynomial system solvers. This thesis makes several contributions to this effort. We propose asymptotically fast algorithms for the core operati ons on which triangular decompositions rely. Complexity results and comparati ve implementation show that these new algorithms provide substantial performance improvements. We present a fundamental software library for polynomial arithmetic in order to support the implementation of high-performance solvers based on triangular decompositions. We investigate strategies for the integration of this library in high-level programming environments where triangular decompositions are usually implemented. We obtain a high performance library combining highly optimized C routines and solving procedures written in the Maple computer algebra system. The experimental result shows that our approaches are very effective, since our code often outperforms pre-existing solvers in a significant manner." } \end{chunk} \index{Li, Xin} \index{Moreno Maza, Marc} \begin{chunk}{axiom.bib} @InProceedings{{Lixx06, author = "Li, Xin and Moreno Maza, Marc", title = "Efficient implementation of polynomial arithmetic in a multiple-level programming environment", booktitle = "Mathematical Software", series = "ICMS 2006", year = "2006", isbn = "978-3-540-38084-9", location = "Spain", pages = "12-23", keywords = "axiomref", paper = "Lixx06.pdf", url = "http://www.math.kobe-u.ac.jp/icms2006/icms2006-video/slides/046.pdf", abstract = "The purpose of this study is to investigate implementation techniques for polynomial arithmetic in a multiple-level programming environment. Indeed, certain polynomial data types and algorithms can further take advantage of the features of lower level languages, such as their specialized data structures or direct access to machine arithmetic. Whereas, other polynomial operations, like Groebner basis over an arbitrary field, are suitable for generic programming in a high-level language. We are interested in the integration of polynomial data type implementations realized at different language levels, such as Lisp, C and Assembly. In particular, we consider situations for which code from different levels can be combined together within the same application in order to achieve high-performance. We have developed implementation techniques in the multiple-level programming environment provided by the computer algebra system AXIOM. For a given algorithm realizing a polynomial operation, available at the user level, we combine the strengths of each language level and the features of a specific machine architecture. Our experimentations show that this allows us to improve performances of this operation in a significant manner." } \end{chunk} \index{Rubey, Martin} \begin{chunk}{axiom.bib} @InProceedings{Rube06, author = "Rubey, Martin", title = "Extented rate, more GFUN", booktitle = "4th Colloquium on mathematics and computer science", series = "DMTCS", year = "2006", location = "Nancy, France", pages = "431-434", paper = "Rube06.pdf", url = "http://mathinfo06.iecn.u-nancy.fr/papers/dmAG431-434.pdf", abstract = "We present a software package that guesses formulas from sequences of, for example, rational numbers or rational functions, given the first few terms. Thereby we extend and complement C. Krattenthaler's program RATE [RATE: a Mathematics guessing machine] and the relevant parts of B. Salvy and P. Zimmermann's GFUN." } \end{chunk} \index{Hebisch, Waldemar} \index{Rubey, Martin} \begin{chunk}{axiom.bib} @article{Hebi10, author = "Hebisch, Waldemar and Rubey, Martin", title = "Extended Rate, more GFUN", year = "2010", url = "https://arxiv.org/abs/math/0702086v2", paper = "Hebi10.pdf", comment = "\newline\refto{package GUESS Guess} \newline\refto{package GUESSAN GuessAlgebraicNumber} \newline\refto{package GUESSF GuessFinite} \newline\refto{package GUESSF1 GuessFiniteFunctions} \newline\refto{package GUESSINT GuessInteger} \newline\refto{package GUESSP GuessPolynomial} \newline\refto{package GUESSUP GuessUnivariatePolynomial}", abstract = "We present a software package that guesses formulae for sequences of, for example, rational numbers or rational functions, given the first few terms. We implement an algorithm due to Bernhard Beckermann and George Labahn, together with some enhancements to render our package efficient. Thus we extend and complement Christian Krattenthaler's program Rate, the parts concerned with guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's qGeneratingFunctions.m." } \end{chunk} \index{Fortuna, E.} \index{Gianni, P.} \index{Luminati, D.} \index{Parenti, P.} \begin{chunk}{axiom.bib} @article{Fort05, author = "Fortuna, E. and Gianni, P. and Luminati, D. and Parenti, P.", title = "The adjacency graph of a real algebraic surface", journal = "Appl. Algebra Eng. Commun. Comput.", volume = "16", number = "5", pages = "271-292", year = "2005", keywords = "axiomref", url = "http://eprints.biblio.unitn.it/788/1/UTM671.pdf", paper = "Fort05.pdf", abstract = "The paper deals with the question of recognizing the mutual positions of the connected components of a non-singular real projective surface$S$in the real projective 3-space. We present an algorithm that answers this question through the computation of the adjacency graph of the surface; it also allows to decide whether each connected component is contractible or not. The algorithm, combined with a previous one returning as an output the topology of the surface, computes a set of data invariant up to ambient-homeomorphism which, though not sufficient to determine the pair$(\mathbb{R}\mathbb{P}^3,S)$, give information about the nature of the surface as an embedded object." } \end{chunk} committed Jun 25, 2016 Commits on Jun 25, 2016 1. Goal: Axiom Literate Programming committed Jun 25, 2016 2. Goal: Axiom Literate Programming committed Jun 25, 2016 3. Goal: Axiom Literate Programming \index{Kredel, Heinz} \begin{chunk}{axiom.bib} @article{Kred08, author = "Kredel, Heinz", title = "On a Java computer algebra system, its performance and applications", journal = "Sci. Comput. Program.", volume = "70", number = "2-3", pages = "185-207", year = "2008", url = "http://ac.els-cdn.com/S0167642307001736/1-s2.0-S0167642307001736-main.pdf", paper = "Kred08.pdf", keywords = "axiomref", abstract = "This paper considers Java as an implementation language for a starting part of a computer algebra library. It describes a design of basic arithmetic and multivariate polynomial interfaces and classes which are then employed in advanced parallel and distributed Groebner base algorithms and applications. The library is type-safe due to its design with Java's generic type parameters and thread-safe using Java's concurrent programming facilities. We report on the performance of the polynomial arithmetic and on applications built upon the core library." } \end{chunk} \index{Kredel, Heinz} \begin{chunk}{axiom.bib} @InProceedings{{Kred07, author = "Kredel, Heinz", title = "Evaluation of a Java computer algebra system", booktitle = "Computer Mathematics, 8th Asian symposium", series = "ASCM 2007", year = "2007", isbn = "978-3-540-87826-1", location = "Singapore", pages = "121-138", keywords = "axiomref", paper = "Kred07.pdf", url = "http://krum.rz.uni-mannheim.de/kredel/oocas-casc2010-slides.pdf", abstract = "This paper evaluates the suitability of Java as an implementation language for the foundations of a computer algebra library. The design of basic arithmetic and multivariate polynomial interfaces and classes have been presented. The library is type-safe due to its design with Java's generic type parameters and thread-safe using Java's concurrent programming facilities. We evaluate some key points of our library and differences to other computer algebra systems." } \end{chunk} \index{Cox, David} \index{Little, John} \index{O'Shea, Donal} \begin{chunk}{axiom.bib} @book{Coxx07, author = "Cox, David and Little, John and O'Shea, Donal", title = "Ideals, varieties and algorithms. An introduction to computational algebraic geometry and commutative algebra", publisher = "Springer", isbn = "978-0-387-35650-1", year = "2007", keywords = "axiomref", abstract = "Around 1980 two new directions in science and technique came together. One was Buchberger’s algorithms in order to handle Groebner bases in an effective way for solving polynomial equations. The second one was the development of the personal computers. This was the starting point of a computational perspective in commutative algebra and algebraic geometry. In 1991 the three authors invented the first edition of their book as an introduction for undergraduates to some interesting ideas in commutative algebra and algebraic geometry with a strong perspective to practical and computational aspects. A second revised edition appeared in 1996. That means from the very beginning the book provides a bridge for the new, computational aspects in the field of commutative algebra and algebraic geometry. To be more precise, the book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations, and elimination theory. Some more spectacular applications are about robotics, automatic geometric theorem proving, and invariants of finite groups. It seems to the reviewer to carry coals to Newcastle for estimating the importance and usefulness of the book. It should be of some interest to ask how many undergraduates have been introduced to algorithmic aspects of commutative algebra and algebraic geometry following the line of the book. The reviewer will be sure that this will continue in the future too. What are the changes to the previous editions? There is a significant shorter proof of the Extension Theorem, see 3.6 in Chapter 3, suggested by A.H.M. Levelt. A major update has been done in Appendix C Computer Algebra Systems''. This concerns in the main the section about MAPLE. Some minor updated information concern the use of AXIOM, CoCoA, Macaulay2, Magma, Mathematica, and SINGULAR. This reflects about the recent developments in Computer Algebra Systems. It encourages an interested reader to more practical exercises. The authors have made changes on over 200 pages to enhance clarity and correctness. Many individuals have reported typographical errors and gave the authors feedback on the earlier editions. The book is well-written. The reviewer guesses that it will become more and more difficult to earn 1 dollar (sponsored by the authors) for every new typographical error as it was the case also with the first and second edition. The reviewer is sure that it will be a excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry." } \end{chunk} \index{Li, Xin} \index{Moreno Maza, Marc} \index{Schost, Eric} \begin{chunk}{axiom.bib} @InProceedings{{Lixx07, author = "Li, Xin and Moreno Maza, Marc and Schost, Eric", title = "Fast arithmetic for triangular sets: from theory to practice", booktitle = "Proc 32nd ISSAC", series = "ISSAC 2007", year = "2007", isbn = "978-1-59593-743-8", location = "Canada", pages = "269-276", keywords = "axiomref", paper = "Lixx09.pdf", url = "http://www.csd.uwo.ca/~moreno/Publications/LiMorenoSchost-ISSAC-2007.pdf", abstract = "We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasi-linear complexity. The main outcome we have in mind is the acceleration of higher-level algorithms, by interfacing our low-level implemention with languages such as AXIOM or Maple. We show the potential for hugh speed-ups, by comparing two AXIOM implementations of vanHoeij and Monagan's modular GCD algorithm." } \end{chunk} \index{Monagan, Michael} \index{Pearce, Roman} \begin{chunk}{axiom.bib} @InProceedings{{Lixx07, author = "Monagan, Michael and Pearce, Roman", title = "Polynomial division using dynamic arrays, heaps, and packed exponent vectors", booktitle = "Computer algebra in scientific computing", series = "CASC 2007", year = "2007", isbn = "978-3-540-75186-1", location = "Bonn", pages = "295-315", keywords = "axiomref", paper = "Mona07.pdf", url = "http://www.cecm.sfu.ca/~rpearcea/sdmp/sdmp\_paper.pdf", abstract = "A common way of implementing multivariate polynomial multiplication and division is to represent polynomials as linked lists of terms sorted in a term ordering and to use repeated merging. This results in poor performance on large sparse polynomials. In this paper we use an auxiliary heap of pointers to reduce the number of monomial comparisons in the worst case while keeping the overall storage linear. We give two variations. In the first, the size of the heap is bounded by the number of terms in the quotient(s). In the second, which is new, the size is bounded by the number of terms in the divisor(s). We use dynamic arrays of terms rather than linked lists to reduce storage allocations and indirect memory references. We pack monomials in the array to reduce storage and to speed up monomial comparisons. We give a new packing for the graded reverse lexicographical ordering. We have implemented the heap algorithms in C with an interface to Maple. For comparison we have also implemented Yan’s geobuckets'' data structure. Our timings demonstrate that heaps of pointers are comparable in speed with geobuckets but use significantly less storage." } \end{chunk} \index{Page, William S.} \begin{chunk}{axiom.bib} @InProceedings{Page07, author = "Page, William S.", title = "Axiom - Open Source Computer Algebra System", booktitle = "Poster ISSAC 2007 Proceedings", series = "ISSAC 2007", year = "2007", volume = "41", number = "3", pages = "114", keywords = "axiomref" } \end{chunk} \index{Pritchard, F. Leon} \index{Sit, William Y.} \begin{chunk}{axiom.bib} @InProceedings{Prit06, author = "Pritchard, F. Leon and Sit, William Y.", title = "On initial value problems for ordinary differential-algebraic equations", booktitle = "Radon Series on Computational and Applied Mathematics", year = "2006", pages = "283-340", isbn = "978-3-11-019323-7", keywords = "axiomref", abstract = "This paper addresses polynomial implicit ODEs in an autonomous context. These ODEs are defined by a system of the form $f_i(z_1,\cdots,z_n,\dot{z_1},\cdots,\dot{z_n})=0,\quad i=1,\cdots,m$ where$f_i$is (for$i=1,\ldots,m$) a polynomial in$2n$variables$(X,P)=(X_1,\ldots,X_n,P_1,\ldots,P_n)$. This covers in particular quasilinear systems, often encountered in applications and defined by polynomials$f_i$in which the total degree in the variables$P$is at most one. The approach is close to the geometrical framework of Rabier and Rheinboldt, profitting from the polynomial form of the system. The authors develop an algorithm for the symbolic computation of the set of consistent initial values via ideal-theoretic results; this is based on a stationary algebraic process of prolongation'', together with the notions of the completion of a given ideal and the algebraic index of the system, defined as the number of steps taken by the process to stabilize. Over- and under-determined systems are also accommodated in their framework." } \end{chunk} \index{Smith, Jacob} \index{Dos Reis, Gabriel} \index{Jarvi, Jaakko} \begin{chunk}{axiom.bib} @InProceedings{Smit07, author = "Smith, Jacob and Dos Reis, Gabriel and Jarvi, Jaakko", title = "Algorithmic differentiation in Axiom", booktitle = "ACM SIGSAM Proceedings", series = "ISSAC 2007", year = "2007", pages = "347-354", keywords = "axiomref", isbn = "978-1-59593-743-8", paper = "Smit07.pdf", keywords = "axiomref", abstract = " This paper describes the design and implementation of an algorithmic differentiation framework in the Axiom computer algebra system. Our implementation works by transformations on Spad programs at the level of the typed abstract syntax tree -- Spad is the language for extending Axiom with libraries. The framework illustrates an algebraic theory of algorithmic differentiation, here only for Spad programs, but we suggest that the theory is general. In particular, if it is possible to define a compositional semantics for programs, we define the exact requirements for when a program can be algorithmically differentiated. This leads to a general algorithmic differentiation system, and is not confined to functions which compute with basic data types, such as floating point numbers." } \end{chunk} committed Jun 25, 2016 4. Goal: Axiom Literate Programming \index{McGettrick, Michael} \begin{chunk}{axiom.bib} @article{Mcge10, author = "McGettrick, Michael", title = "One dimensional quantum walks with memory", journal = "Quantum Inf. Comput.", volume = "10", number = "5-6", pages = "509-524", year = "2010", keywords = "axiomref", paper = "Mcge10.pdf", url = "https://arxiv.org/pdf/0911.1653v2.pdf", abstract = "We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov chain is of order 2. This corresponds to the walk having a memory of one previous step. We derive the amplitudes and probabilities for these walks, and point out how they differ from bot classical random walks, and quantum walks without memory." } \end{chunk} \index{Lecerf, Gr{\'e}goire} \begin{chunk}{axiom.bib} @InProceedings{{Lece10, author = "Lecerf, Gregoire", title = "Mathemagix: toward large scale programming for symbolic and certified numeric computations", booktitle = "Mathematical software", series = "ICMS 2010", year = "2010", isbn = "978-3-642-15581-9", location = "Berlin", pages = "329-332", keywords = "axiomref", abstract = "Coordinated by Joris van der Hoeven from the 90's, the Mathemagix project aims at the design of a scientific programming language for symbolic and certified numeric algorithms. This language can be compiled and interpreted, and it features a strong type system with classes and categories. Several C++ libraries are also being developed, mainly with Bernard Mourrain and Philippe Trebuchet, for the elementary operations with polynomials, power series and matrices, with a special care towards efficiency and numeric stability. In my talk I will give an overview of the language, of the design and the contents of the C++ libraries, and I will illustrate possibilities offered for certified numeric computations with balls and intervals." } \end{chunk} \index{Roanes-Lozano, Eugenio} \index{val Labeke, Nicolas} \index{Roanes-Macias, Eugenio} \begin{chunk}{axiom.bib} @article{Roan10, author = "Roanes-Lozano, Eugenio and val Labeke, Nicolas and Roanes-Macias, Eugenio", title = "Connecting the 3D DGS Calques3D with the CAS Maple", journal = "Math. Comput. Simul.", volume = "80", number = "6", pages = "1153-1176", year = "2010", keywords = "axiomref", url = "http://nvl.calques3d.org/publications/2010.MatCom.Connecting.pdf", paper = "Roan10.pdf", abstract = "Many (2D) Dynamic Geometry Systems (DGSs) are able to export numeric coordinates and equations with numeric coefficients to Computer Algebra Systems (CASs). Moreoever, different approaches and systems that linke (2D) DGSs with CASs, so that symbolic coordinates and equations with symbolic coefficients can be exported from the DGS to the CAS, already exist. Although the 2D DGS Calques3D can export numeric coordinates and equations with numeric coefficients to Maple and Mathematica, it cannot export symbolic coordinates and equations with symbolic coefficients. A connetion between the 3D DGS Calques3D and the CAS Maple, that can handle symbolic coordinates and equations with symbolic coefficients, is presented here. Its main interest is to provide a convenient time-saving way to explore problems and directly obtain both algebraic and numeric data when dealing with a 3D extension of ruler and compass geometry''. This link has not only educational purposes but mathematical ones, like mechanical theorem proving in geometry, geometry discovery (hypothesis completion), geometric loci finding -- As far as we know, there is no comparable symbolic'' link in the 3D case, except the prototype 3D-LD (restricted to determining algebraic surfaces as geometric loci)." } \end{chunk} \index{Bradford, Russell} \index{Davenport, James H.} \index{Sangwin, Christopher J.} \begin{chunk}{axiom.bib} @InProceedings{Brad09, author = "Bradford, Russell and Davenport, James H. and Sangwin, Christopher J.", title = "A comparison of equality in computer algebra and correctness in mathematical pedagogy", year = "2009", booktitle = "Intelligent Computer Mathematics, 16th symposium", series = "Calculemus 2009", pages = "75-89", isbn = "978-3-642-02613-3", url = "http://opus.bath.ac.uk/15188/1/RJBJHDCJSv2.pdf", paper = "Brad09.pdf", keywords = "axiomref", abstract = "How do we recognize when an answer is right''? This is a question that has bedevilled the use of computer systems in mathematics (as opposed to arithmetic) ever since their introduction. A computer system can certainly say that some answers are definitely wrong, in the sense that they are provably not an answer to the question posed. However, an answer can be mathematically right without being pedagogically right. Here we explore the differences and show that, despite the apparent distinction, it is possible to make many of the differences amenable to formal treatment, by asking under which congruence is the pupil's answer equal to the teacher's?''." } \end{chunk} \index{Li, Xin} \index{Moreno Maza, Marc} \index{Schost, Eric} \begin{chunk}{axiom.bib} @article{Lixx09, author = "Li, Xin and Moreno Maza, Marc and Schost, Eric", title = "Fast arithmetic for triangular sets: from theory to practice", journal = "J. Symb. Comput.", volume = "44", number = "7", pages = "891-907", year = "2009", keywords = "axiomref", url = "http://www.csd.uwo.ca/~moreno/Publications/LiMorenoSchost-ISSAC-2007.pdf", paper = "Lixx09.pdf", abstract = "We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasi-linear complexity. The main outcome we have in mind is the acceleration of higher-level algorithms, by interfacing our low-level implemention with languages such as AXIOM or Maple. We show the potential for hugh speed-ups, by comparing two AXIOM implementations of vanHoeij and Monagan's modular GCD algorithm." } \end{chunk} \index{Rioboo, Renaud} \begin{chunk}{axiom.bib} @article{Riob09, author = "Rioboo, Renaud", title = "Invariants for the FoCaL language", journal = "Ann. Math. Artif. Intell.", volume = "56", number = "3-4", pages = "273-296", year = "2009", keywords = "axiomref", abstract = "We present a FoCaL formalization for quotient structures which are common in mathematics. We first present a framework for stating invariant properties of the data manipulated by running programs. A notion of equivalence relation is then encoded for the FoCaL library. It is implemented through projections functions, this enables us to provide canonical representations which are commonly used in Computer Algebra but seldom formally described. We further provide a FoCaL formalization for the code used inside the library for modular arithmetic through the certification of quotient groups and quotient rings which are involved in the model. We finally instantiate our framework to provide a trusted replacement of the existing FoCaL library." } \end{chunk} \index{Rioboo, Renaud} \begin{chunk}{axiom.bib} @misc{Riobxx, author = "Rioboo, Renaud", title = "Cylindrical Algebraic Decomposition class notes", url = "http://people.bath.ac.uk/masjhd/TRITA.pdf", paper = "Riobxx.pdf", year = "2016", abstract = "This report describes techniques for resolving systems of polynomial equations and inequalities. The general technique is {\sl cylindrical algebraic decomposition}, which decomposes space into a number of regions, on each of which the equations and inequalities have the same sign. Most of the report is spent describing the algebraic and algorithmic pre-requisites (resultants, algebraic numbers, Sturm sequences, etc.), and then describing the method, first in two dimensions and then in an arbitrary number of dimensions" } \end{chunk} \index{Akbar Hussain, D.M.} \index{Haq, Shaiq A.} \index{Khan, Zafar Ullah} \index{Ahmed, Zaki} \begin{chunk}{axiom.bib} @article{Akba08, author = "Akbar Hussain, D.M. and Haq, Shaiq A. and Khan, Zafar Ullah and Ahmed, Zaki", title = "Simple object oriented designed computer algebra system", journal = "J. Comput. Mathods Sci. Eng.", volume = "8", number = "3", pages = "195-211", year = "2008", keywords = "axiomref", abstract = "Computer Algebra System (CAS) is a software program that facilitates symbolic mathematics. The core functionality of a typical CAS is manipulation of mathematical expressions in symbolic forms. The expressions manipulated by CAS normally include polynomials in multiple variables, standart trigonometric and exponential functions, various special functions for example gamma, zeta, Bessel, etc. and also arbitrary functions like derivatives, integrals, sums, and products of expressions. Our implementation of a CAS tool provides an object orienged design framework. The system is portable to other platforms and highly scalable. The other key features include a very simple and interactive user GUI support for a formula editor, making it a self contained system, additionally the formula editor provides a real-time syntax checking for expressions." } \end{chunk} \index{Joyner, David} \begin{chunk}{axiom.bib} @article{Joyn08, author = "Joyner, David", title = "Open source computer algebra systems: Axiom", journal = "ACM Commun. Comput. Algebra", volume = "42", number = "1-2", pages = "39-47", year = "2008", keywords = "axiomref", abstract = "This survey will look at Axiom, a free and very powerful computer algebra system available. It is a general purpose CAS useful for symbolic computation, research, and the development of new mathematical algorithms. Axiom is similar in some ways to Maxima, covered in the survey, but different in many ways as well. Axiom, Maxima, and SAGE, are the largest of the general-purpose open-source CASs." } \end{chunk} committed Jun 25, 2016 5. Goal: Axiom Test Code committed Jun 25, 2016 6. Goal: Axiom Literate Programming Collect algebra references in the bibliography \index{Carette, Jacques} \index{Kiselyov, Oleg} \begin{chunk}{axiom.bib} @article{Care11, article = "Carette, Jacques and Kiselyov, Oleg", title = "Multi-stage programming with functors and monads: eliminating abstraction overhead from generic code", journal = "Sci. Comput. Program", volume = "76", number = "5", pages = "349-375", year = "2011", paper = "Care11.pdf", url = "http://www.cas.mcmaster.ca/~carette/metamonads/metamonads.pdf", keywords = "axiomref", abstract = "We use multi-stage programming, monads and OCaml's advanced module system to demonstrate how to eliminate all abstraction overhead from generic programs, while avoiding any inspection of the resulting code. We demonstrate this clearly with Gaussian Elimination as a representative family of symbolic and numeric algorithms. We parameterize our code to a great extent -- over domain, input and permutation matrix representations, determinant and rank tracking, pivoting policies, result types, etc. -- at no run-time cost. Because the resulting code is generated just right and not changed afterward, MetoOCaml guarantees that the generated code is well-typed. We further demonstrate that various abstraction parameters (aspects) can be made orthogonal and compositional, even in the presence of name-generation for temporaries, and interleaving'' of aspects. We also show how to encode some domain-specific knowledge so that clearly wrong'' compositions can be rejected at or before generation time, rather than during the compilation or running of the generated code." } \end{chunk} \index{Rojas-Bruna, Carlos} \begin{chunk}{axiom.bib} \article{Roja13, author = "Rojas-Bruna, Carlos", title = "Trace forms and ideals on commutative algebras satisfying an identity of degree four", journal = "Rocky Mt. J. Math.", year = "2013", volume = "43", number = "4", pages = "1325-1336", keywords = "axiomref", abstract = "This paper deals with the variety of commutative algebras satsifying the identity $((xy)z)t - ((xy)t)z + ((yt)x)z - ((yt)z)x + ((yz)t)x - ((yz)x)t = 0$ These algebras appeared in the classification of the degree four identities in Carini et al. We prove the existence of a trace form. Moreover, if we assume the existence of degenerate trace form, the$A$satisfies the identity$((yx)x)x=y((xx)x)$, a generalization of right-alternativity. Finally we prove that$Ass[A]$and$N(A)$are ideals in these algebras." } \end{chunk} \index{Dos Reis, Gabriel} \begin{chunk}{axiom.bib} \article{Reis12, author = "Dos Reis, Gabriel", title = "A System for Axiomatic Programming", journal = "Proc. Conf. on Intelligent Computer Mathematics", publisher = "Springer", year = "2012", url = "http://www.axiomatics.org/~gdr/liz/cicm-2012.pdf", paper = "Reis12.pdf", keywords = "axiomref", abstract = " We present the design and implementation of a system for axiomatic programming, and its application to mathematical software construction. Key novelties include a direct support for user-defined axioms establishing local equality between types, and overload resolution based on equational theories and user-defined local axioms. We illustrate uses of axioms, and their organization into concepts, in structured generic programming as practiced in computational mathematical systems." } \end{chunk} \index{Dos Reis, Gabriel} \index{Matthews, David} \index{Li, Yue} \begin{chunk}{axiom.bib} @article{Reis11, author = "Dos Reis, Gabriel", title = "Retargeting OpenAxiom to Poly/ML: towards an integrated proof assistants and computer algebra system framework", journal = "Intelligent computer mathematics (MKM 2011)", year = "2011", isbn = "978-3-642-22672-4", paper = "Reis11.pdf", url = "https://www.semanticscholar.org/paper/Retargeting-OpenAxiom-to-PolyML-Towards-an-Reis-Matthews/4ce5d85ea8424ced82d", keywords = "axiomref", abstract = "This paper presents an ongoing effort to integrate the AXIOM family of computer algebra systems with Poly/ML-based 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} \index{Kredel, Heinz} \begin{chunk}{axiom.bib} @book{Kred11, author = "Kredel, Heinz", title = "Unique factorization domains in the Java computer algebra system", year = "2011", booktitle = "Automated deduction in geometry (ADG 2008)", pages = "86-115", isbn = "978-3-642-21045-7", keywords = "axiomref", abstract = "This paper describes the implementation of recursive algorithms in unique factorization domains, namely multivariate polynomial greatest common divisors (gcd) and factorization into irreducible parts in the Java computer algebra library (JAS). The implementation of gcds, resultants and factorization is part of the essential building blocks for any computation in algebraic geometry, in particular in automated deduction in geometry. There are various implementations of these algorithms in procedural programming languages. Our aim is an implementation in a modern object oriented language with generic data types, as it is provided by Java programming language. We exemplify that the type design and implementation of JAS is suitable for the implementation of several greatest common divisor algorithms and factorization of multivariate polynomials. Due to the design we can employ this package in very general settings not commonly seen in other computer algebra systems. As for example, in the coefficient arithmetic for advanced Groebner basis computations like in polynomial rings over rational function fields or (finite, commutative) regular rings. The new package provides factory methods for the selection of one of the several implementations for non experts. Further we introduce a parallel proxy for gcd implementations which runs different implementations concurrently." } \end{chunk} \index{Kredel, Heinz} \index{Jolly, Raphael} \begin{chunk}{axiom.bib} @InProceedings{{Kred11a, author = "Kredel, Heinz and Jolly, Raphael", title = "Algebraic structures as typed objects", booktitle = "Proc. 13th International Workshop", series = "CASC 2011", year = "2011", isbn = "978-3-642-23567-2", location = "Berlin", pages = "294-308", keywords = "axiomref", paper = "Kred11a.pdf", url = "http://krum.rz.uni-mannheim.de/kredel/to-cas-casc2011-slides.pdf", abstract = "Following the research direction of strongly typed, generic, object oriented computer algebra software, we examine the modeling of algebraic structures as typed objects in this paper. We discuss the design and implementation of algebraic and transcendental extension fields together with the modeling of real algebraic and complex algebraic extension fields. We will show that the modeling of the relation between algebraic and real algebraic extension fields using the delegation design concept has advantages over the modeling as sub-types using sub-class implementation. We further present a summary of design problems, which we have encountered so far with our implementation in Java and present possbile solutions in Scala." } \end{chunk} \index{Spitters, Bas} \index{van der Weegen, Eelis} \begin{chunk}{axiom.bib} @article{Spit11, author = "Spitters, Bas and van der Weegen, Eelis", title = "Type classes for mathematics in type theory", journal = "Math. Struct. Comput. Sci.", volume = "21", number = "4", pages = "795-825", year = "2011", keywords = "axiomref", url = "https://arxiv.org/pdf/1102.1323.pdf", paper = "Spit11.pdf", abstract = "The introduction of first-class type classes in the Coq system calls for a re-examination of the basic interfaces used for mathematical formalisation in type theory. We present a new set of type classes for mathematics and take full advantage of their unique features to make practical a particularly flexible approach that was formerly thought to be infeasible. Thus, we address traditional proof engineering challenges as well as new ones resulting from our ambition to build upon this development a library of constructive analysis in which any abstraction penalties inhibiting efficient computation are reduced to a minimum. The basis of our development consists of type classes representing a standard algebraic hierarchy, as well as portions of category theory and universal algebra. On this foundation, we build a set of mathematically sound abstract interfaces for different kinds of numbers, succinctly expressed using categorical language and universal algebra constructions. Strategic use of type classes lets us support these high-level theory-friendly definitions, while still enabling efficient implementations unhindered by gratuitous indirection, conversion or projection. Algebra thrives on the interplay between syntax and semantics. The Prolog-like abilities of type class instance resolution allow us to conveniently define a quote function, thus facilitating the use of reflective techniques." } \end{chunk} \index{Kredel, Heinz} \index{Jolly, Raphael} \begin{chunk}{axiom.bib} @InProceedings{{Kred10, author = "Kredel, Heinz and Jolly, Raphael", title = "Generic, type-safe and object oriented computer algebra software", booktitle = "Proc. 12th International Workshop", series = "CASC 2010", year = "2010", isbn = "978-3-642-15273-3", location = "Berlin", pages = "162-177", keywords = "axiomref", paper = "Kred10.pdf", url = "http://krum.rz.uni-mannheim.de/kredel/oocas-casc2010-slides.pdf", abstract = "Advances in computer science, in particular object oriented programming, and software engineering have had little practical impact on computer algebra systems in the last 30 years. The software design of existing systems is still dominated by ad-hoc memory management, weakly typed algorithm libraries and proprietary domain specific interactive expression interpreters. We discuss a modular approach to computer algebra software: usage of state-of-the-art memory management and run-time systems (e.g. JVM) usage of strongly typed, generic, object oriented programming languages (e.g. Java) and usage of general purpose, dynamic interactive expression interpreters (e.g. Python). To illuatrate the workability of this approach, we have implemented and studied computer algebra systems in Java and Scala. In this paper we report on the current state of this work by presenting new examples." } \end{chunk} \index{Lecerf, Gr{\'e}goire} \begin{chunk}{axiom.bib} @InProceedings{{Lece10, author = "Lecerf, Gregoire", title = "Mathemagix: toward large scale programming for symbolic and certified numeric computations", booktitle = "Mathematical software", series = "ICMS 2010", year = "2010", isbn = "978-3-642-15581-9", location = "Berlin", pages = "329-332", keywords = "axiomref", abstract = "Coordinated by Joris van der Hoeven from the 90's, the Mathemagix project aims at the design of a scientific programming language for symbolic and certified numeric algorithms. This language can be compiled and interpreted, and it features a strong type system with classes and categories. Several C++ libraries are also being developed, mainly with Bernard Mourrain and Philippe Trebuchet, for the elementary operations with polynomials, power series and matrices, with a special care towards efficiency and numeric stability. In my talk I will give an overview of the language, of the design and the contents of the C++ libraries, and I will illustrate possibilities offered for certified numeric computations with balls and intervals." } \end{chunk} committed Jun 25, 2016 Commits on Jun 24, 2016 1. Goal: Axiom Literate Programming Collect algebra references in the bibliography \index{Rothstein, Michael} \index{Caviness, Bob F.} \begin{chunk}{axiom.bib} @article{Ro76a, author = "Rothstein, Michael and Caviness, Bob F.", title = "A structure theorem for exponential and primitive functions: a preliminary report", journal = "ACM Sigsam Bulletin", volume = "10", number = "4", year = "1976", paper = "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$F_{i-1}=F(t_1,\ldots,t_{i-1})$, is an exponential of an element in$F_{i-1}$, or is an integral of an element in$F_{i-1}$. 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} \index{Singer, Michael F.} \index{Saunders, B. David} \index{Caviness, Bob F.} \begin{chunk}{axiom.bib} @article{Sing85, author = "Singer, Michael F. and Saunders, B. David and Caviness, Bob F.", title = "An extension of Liouville's theorem on integration in finite terms", journal = "SIAM J. of Comp.", volume = "14", pages = "965-990", year = "1985", url = "http://www4.ncsu.edu/~singer/papers/singer_saunders_caviness.pdf", paper = "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} \index{Buchberger, Bruno} \index{Caviness, Bob F.} \begin{chunk}{axiom.bib} @book{Buch85, author = "Buchberger, Bruno and Caviness, Bob F. (eds)", title = "Lecture Notes in Computer Science, Vol 204", isbn = "0-387-15983-5 (vol 1), 0-387-15984-3 (vol 2)", year = "1985", month = "April", publisher = "Springer-Verlag, Berlin, Germany", keywords = "axiomref" } \end{chunk} \index{Kokol-voljc, Vlasta} \index{Kutzler, Bernhard} \begin{chunk}{axiom.bib} @misc{ACA00, authors = "Kokol-voljc, Vlasta and Kutzler, Bernhard", title = "Computer Algebra Meets Education", keywords = "axiomref", conference = "Sessions of ACA2000", abstract = "Education has become one of the fastest growing application areas for computers in general and computer algebra in particular. Computer algebra tools such as TI-92/89, Derive, Mathematica, Maple, Axiom, Reduce, Macsyma, or Mupad make powerful teaching tools in mathematics, physics, chemistry, biology, economy. The goal of this session is to exchange ideas and experiences, to hear about classroom experiments, and to discuss all issues related to the use of computer algebra tools in the classroom (such as assessment, change of curricula, new support material, ...)" } \end{chunk} \index{van Leeuwen, Andr\'e M.A.} \begin{chunk}{axiom.bib} @article{Leeu94, author = "van Leeuwen, Andre M.A.", title = "LiE, A software package for Lie group computations", journal = "Euromath Bulletin", volume = "1", number = "2", year = "1994", keywords = "axiomref", paper = "Leeu94.pdf", abstract = "A description is given of LiE, a specialized computer algebra package for computations concerning Lie groups and algebras, and their representations. The functionality of the package is demonstrated by sample computations, and the structure of the program and the algorithms implemented in it are discussed." } \end{chunk} \index{Hoarau, Emma} \index{David, Claire} \begin{chunk}{axiom.bib} @article{Hoar08, author = "Hoarau, Emma and David, Claire", title = "Lie group computation of finite difference schemes", year = "2008", journal = "math.NA", url = "http://arxiv.org/pdf/math/0611895.pdf", paper = "Hoar08.pdf", keywords = "axiomref", abstract = "A Mathematica based program has been elaborated in order to determine the symmetry group of a finite difference equation. The package provides functions which enabel use to solve the determining equations of the related Lie group." } \end{chunk} \begin{chunk}{axiom.bib} @misc{Ency16, author = "Unknown", title = "Encyclopedia of Mathematics", url = "https://www.encyclopediaofmath.org/index.php/Computer\_algebra\_package", keywords = "axiomref" } \end{chunk} \begin{chunk}{axiom.bib} @misc{Swmath, author = "Unknown", title = "Axiom", url = "https://www.swmath.org/software/63", keywords = "axiomref", abstract = "Axiom is a general purpose Computer Algebra System (CAS). It is useful for research and developement of mathematical algorithms. It defines a strongly typed, mathematically correct type hierarchy. It has a programming language and a built-in compiler." } \end{chunk} \index{Heras, Jonathan} \index{Martin-Mateos, Franciso Jesus} \index{Pascual, Vico} @article{Hera15, author = "Heras, Jonathan and Martin-Mateos, Franciso Jesus and Pascual, Vico", title = "Modelling algebraic structures and morphisms in ACL2", journal = "Appl. Algebra Eng. Commun. Comput.", volume = "26", number = "3", pages = "277-303", year = "2015", keywords = "axiomref", abstract = "In this paper, we present how algebraic structures and morphisms can be modelled in the ACL2 theorem prover. Namely, we illustrate a methodology for implementing a set of tools that facilitates the formalisations related to algebraic structures -- as a result, an algebraic hierarchy ranging from setoids to vector spaces has been developed. The resultant tools can be used to simplify the development of generic theories about algebraic structures. In particular, the benefits of using the tools presented in this paper, compared to a from-scratch approach, are especially relevant when working with complex mathematical structures; for example, the structures employed in Algebraic Topology. This work shows that ACL2 can be a suitable tool for formalising algebraic concepts coming, for instance, from computer algebra systems." } \end{chunk} \index{Heras, Jonathan} \index{Martin-Mateos, Franciso Jesus} \index{Pascual, Vico} \begin{chunk}{axiom.bib} @misc{Hera16, author = "Heras, Jonathan and Martin-Mateos, Franciso Jesus and Pascual, Vico", title = "A Hierarchy of Mathematical Structures in ACL2", keywords = "axiomref", paper = "Hera16.pdf", url = "http://staff.computing.dundee.ac.uk/jheras/papers/ahomsia.pdf", abstract = "In this paper, we present a methodology which allows one to deal with {\sl mathematical structures} in the ACL2 theorem prover. Namely, we cope with the representation of mathematical structures, the certification that an object fulfills the axioms characterizing an algebraic structure and the generation of generic theories about concrete structures. As a by-product, an {\sl ACL2 algebraic hierarchy} has been obtained. Our framework has been tested with the definition of {\sl homology groups}, an example coming from Homological Algebra which involves several notions related to Universal Algebra. The method presented here, when compared to a from-scratch approach, is preferred when working with complex mathematical structures; for instance, the ones coming from Algebraic Topology. The final aim of this work is the verification of Computer Algebra systems, a field where our hierarchy fits better than the ones developed in other systems." } \end{chunk} \begin{chunk}{axiom.bib} @misc{ORMS, author = "Unknown", title = "Oberwolfach References on Mathematical Software", url = "http://orms.mfo.de/project?id=234", keywords = "axiomref" } \end{chunk} \index{Ballarin, Clemens} \begin{chunk}{axiom.bib} @article{Ball14, author = "Ballarin, Clemens", title = "Locales: a module system for mathematical theories", journal = "J. Autom. Reasoning", volume = "52", number = "2", pages = "123-153", year = "2014", keywords = "axiomref", url = "http://www21.in.tum.de/~ballarin/publications/jar2013.pdf", paper = "Ball14.pdf", abstract = "Locales are a module system for managing theory hierarchies in a theorem prover through theory interpretation. They are available for the theorem prover Isabelle. In this paper, their semantics is defined in terms of local theories and morphisms. Locales aim at providing flexible means of extension and reuse. Theory modules (which are called locales) may be extended by definitions and theorems. Interpretation of Isabelle's global theories and proof contexts is possible via morphisms. Even the locale hierarchy may be changed if declared relations between locales do not adequately reflect logical relations, which are implied by the locales' specifications. By discussing their design and relating it to more commonly known structuring mechanisms of programming languages and provers, locales are made accessible to a wider audience beyond the users of Isabelle. The discussed mechanisms include ML-style functors, type classes and mixins (the latter are found in modern object-oriented languages)." } \end{chunk} \index{van der Hoeven, Joris} \begin{chunk}{axiom.bib} @article{Hoev12, author = "van der Hoeven, Joris", title = "Overview of the Mathemagix type system", journal = "ASCM 2102", year = "2012", keywords = "axiomref", url = "http://www.texmacs.org/joris/mmxtyping/mmxtyping.pdf", paper = "Hoev12.pdf", abstract = "The goal of the {\tt MATHEMAGIX} project is to develop a new and free software for computer algebra and computer analysis, base on a strongly typed and compiled language. In this paper, we focus on the nderlying type system of this language, which allows for heavy overloading, including parameterized overloading with parameters in so called categories''. The exposition is informal and aims at giving the reader an overview of the main concepts, ideas and differences with existing languages. In a forthcoming paer, we intend to describe the formal semantics of the type system in more detail." } \end{chunk} committed Jun 24, 2016 2. Goal: Axiom Literate Programming Collect algebra references in the bibliography \index{Arnault, Francois} \begin{chunk}{axiom.bib} @Article{Arna95, author = "Arnault, Francois", title = "Constructing Carmichael numbers which are strong pseudoprimes to several bases", year = "1995", journal = "Journal of Symbolic Computation", volume = "20", number = "2", pages = "151-161", paper = "Arna95.pdf", url = "http://ac.els-cdn.com/S0747717185710425/1-s2.0-S0747717185710425-main.pdf", keywords = "axiomref", comment = "\newline\refto{package PRIMES IntegerPrimesPackage}", abstract = "We describe here a method of constructing Carmichael numbers which are strong pseudoprimes to some set of prime bases. We apply it to find composite numbers which are found to be prime by the Rabin-Miller test of packages as Axiom or Maple. We also use a variation of this method to construct strong Lucas pseudoprimes with respect to several pairs of parameters." } \index{Davenport, James H.} \begin{chunk}{axiom.bib} @article{Dave92, author = "Davenport, James H.", title = "Primality Testing Revisited", url = "http://staff.bath.ac.uk/masjhd/ISSACs/ISSAC1992.pdf", paper = "Dave92.pdf", report = "Technical Report TR2/93 Numerical Algorithms Group, Inc", keywords = "axiomref", comment = "\newline\refto{package PRIMES IntegerPrimesPackage}", abstract = "Rabin's algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm." } \end{chunk} \index{Jaeschke, Gerhard} \begin{chunk}{axiom.bib} @article{Jaes93, author = "Jaeschke, Gerhard", title = "On String Pseudoprimes to Several Bases", journal = "Mathematics of Computation", volume = "61", number = "204", year = "1993", pages = "915-926", paper = "Jaes93.pdf", keywords = "axiomref", comment = "\newline\refto{package PRIMES IntegerPrimesPackage}", abstract = "With$\psi_k$denoting the smallest strong pseudoprime to all of the first$k$primes taken as bases we determine the exact values for$\psi_5$,$\psi_6$,$\psi_7$,$\psi_8$, and give upper bounds for$\psi_9$,$\psi_{10}$,$\psi_{11}$. We discuss the methods and underlying facts for obtaining these results" } \end{chunk} committed Jun 24, 2016 3. Goal: Axiom Literate Programming Collect algebra references in the bibliography \begin{chunk}{axiom.bib} @misc{Sympy, keywords = "axiomref", url = "https://github.com/sympy/sympy/wiki/SymPy-vs.-Axiom" } \end{chunk} \begin{chunk}{axiom.bib} @misc{America, keywords = "axiomref", url = "http://america.pink/axiom-computer-algebra-system_526647.html" } \end{chunk} \index{Hereman, Willy} \begin{chunk}{axiom.bib} @misc{Here96, author = "Hereman, Willy", title = "The Incredible World of Symbolic Mathematics A Review of Computer Algebra Systems", year = "1996", keywords = "axiomref", url = "https://inside.mines.edu/~whereman/papers/Hereman-PhysicsWorld-9-March1996.pdf", paper = "Here96.pdf" } \end{chunk} \index{Betten, Anton} \index{Kohnert, Axel} \index{Laue, Reinhard} \index{Wassermann, Alfred} \begin{chunk}{axiom.bib} @book{Bett99, author = "Betten, Anton and Kohnert, Axel and Laue, Reinhard and Wassermann, Alfred", title = "Algebraic Combinatorics and Applications", year = "1999", publisher = "Springer", isbn = "978-3-540-41110-9", keywords = "axiomref" } \end{chunk} \index{Beebe, Nelson H. F.} \begin{chunk}{axiom.bib} @misc{Beeb14, author = "Beebe, Nelson H. F.", title = "A Bibliography of Publications about the AXIOM (formerly, Scratchpad) Symbolic Algebra Language", year = "2014", url = "http://www.netlib.org/bibnet/journals/axiom.ps.gz", paper = "Beeb14.pdf" } \end{chunk} \begin{chunk}{axiom.bib} @misc{ACM89, author = "ACM", title = "Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, ISSAC '89" year = "1989", isbn = "0-89791-325-6", keywords = "axiomref", isbn = "0-89791-325-6", url = "http://doi.acm.org/10.1145/74540.74567", doi = "10.1145/74540.74567", acmid = "74567", publisher = "ACM Press", address = "New York, NY, USA" } \end{chunk} \begin{chunk}{axiom.bib} @misc{ACM94, author = "ACM", title = "Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, ISSAC '94" year = "1994", isbn = "0-89791-638-7", keywords = "axiomref", publisher = "ACM Press", address = "New York, NY, USA" } \end{chunk} \index{Andrews, George E.} \begin{chunk}{axiom.bib} @InProceedings{Andr84, author = "Andrews, George E.", title = "Ramanujan and SCRATCHPAD", booktitle = "Proc. of 1984 MACSYM Users' Conference, July 1984", year = "1984", pages = "383-??", keywords = "axiomref" } \end{chunk} \index{Burge, William H.} \begin{chunk}{axiom.bib} @InProceedings{Burg91, author = "Burge, W.H.", title = "Scratchpad and the Rogers-Ramanujan identities", booktitle = "Proc. ISSAC 1991", year = "1991", pages = "189-190", keywords = "axiomref", paper = "Burg91.pdf", abstract = "This note sketches the part played by Scratchpad in obtaining new proofs of Euler's theorem and the Rogers-Ramanujan Identities." } \end{chunk} \index{Andrews, George E.} \begin{chunk}{axiom.bib} @InProceedings{Andr88, author = "Andrews, G. E.", title = "Application of Scratchpad to problems in special functions and combinatorics", booktitle = "Trends in Computer Algebra", year = "1988", isbn = "3-540-18928-9", keywords = "axiomref", pages = "158-??" } \end{chunk} \begin{chunk}{axiom.bib} @book{Anon91, author = "Anonymous", title = "Challenges of a Changing World (2 Volumes)", publisher = "American Society for Engineering Education", year = "1991, keywords = "axiomref" } \end{chunk} \begin{chunk}{axiom.bib} @book{Anon95, author = "Anonymous", title = "Zeitschrift fur Angewandte Mathematik und Physik, 75 (suppl. 2)", keywords = "axiomref", year = "1995", issn = "0044-2267" } \end{chunk} \index{Arnault, Francois} \begin{chunk}{axiom.bib} @Article{Arna95, author = "Arnault, Francois", title = "Constructing Carmichael numbers which are strong pseudoprimes to several bases", year = "1995", journal = "Journal of Symbolic Computation", volume = "20", number = "2", pages = "151-161", paper = "Arna95.pdf", url = "http://ac.els-cdn.com/S0747717185710425/1-s2.0-S0747717185710425-main.pdf", keywords = "axiomref", abstract = "We describe here a method of constructing Carmichael numbers which are strong pseudoprimes to some set of prime bases. We apply it to find composite numbers which are found to be prime by the Rabin-Miller test of packages as Axiom or Maple. We also use a variation of this method to construct strong Lucas pseudoprimes with respect to several pairs of parameters." } committed Jun 24, 2016 4. Goal: Axiom Literate Programming Collect algebra references in the bibliography \index{Lambe, Larry A.} \begin{chunk}{axiom.bib} @article{Lamb92, author = "Lambe, Larry", title = "Next Generation Computer Algebra Systems AXIOM and the Scratchpad Concept: Applications to Research in Algebra", publisher = "21st Nordic Congress of Mathematicians", year = "1992", paper = "Lamb92.pdf", url = "http://axiom-wiki.newsynthesis.org/public/refs/axiom-21cong.pdf", keywords = "axiomref", comment = "\newline\refto{category FMCAT FreeModuleCat}", abstract = "One way in which mathematicians deal with infinite amounts of data is symbolic representation. A simple example is the quadratic equation $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ 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} committed Jun 24, 2016 5. Goal: Axiom Literate Programming Collect algebra references in the bibliography \index{Aubry, Phillippe} \index{Maza, Marc Moreno} \begin{chunk}{axiom.bib} @article{Aubr96, author = "Aubry, Phillippe and Maza, Marc Moreno", title = "Triangular Sets for Solving Polynomial Systems: a Comparison of Four Methods", url = "http://www.lip6.fr/lip6/reports/1997/lip6.1997.009.ps.gz", paper = "Aubr96.pdf", comment = "\newline\refto{package RSDCMPK RegularSetDecompositionPackage}", 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." } \end{chunk} committed Jun 24, 2016 6. Goal: Axiom Literate Programming Collect algebra references in the bibliography \index{Lazard, Daniel} \begin{chunk}{axiom.bib} @article{Laza91, author = "Lazard, Daniel", title = "A new method for solving algebraic systems of positive dimension", journal = "Discrete. Applied. Mathematics", volume = "33", year = "1991", pages = "147-160", paper = "Laza91.pdf", comment = "\newline\refto{category NTSCAT NormalizedTriangularSetCategory}", abstract = "A new algorithm is presented for solving algebraic systems of equations, which is designed from the structure which is wanted for the result. This algorithm is not yet implemented; thus technical details and proofs are omitted, for emphasising on the relation between the algorithm design and a good representation of the result. The algorithm is based on a new theorem of decomposition for algebraic varieties." } \end{chunk} \index{Maza, Marc Moreno} \index{Rioboo, Renaud} \begin{chunk}{axiom.bib} @article{Maza95, author = "Maza, Marc Moreno and Rioboo, Renaud", title = "Polynomial Gcd Computations over Towers of Algebraic Extensions", year = "1995", journal = "Proceedings of AAECC11", keywords = "axiomref", paper = "Maza95.pdf", comment = "\newline\refto{category NTSCAT NormalizedTriangularSetCategory}", abstract = "Some methods for polynomial system solving require efficient techniques for computing univariate polynomial gcd over algebraic extensions of a field. Currently used techniques compute {\sl generic} univariate polynomial gcd before {\sl specializing} the result using algebraic relations in the ring of coefficients. This strategy generates very big intermediate data and fails for many problems. We present here a new approach which takes permanently into account those algebraic relations. It is based on a property of subresultant remainder sequences and leads to a great increase of the speed of computations and thus the size of accessible systems." } \end{chunk} \index{Maza, Marc Moreno} \begin{chunk}{axiom.bib} @phdthesis{Maza97, author = "Maza, Marc Moreno", title = "Calculs de pgcd au-dessus des tours d'extensions simples et resolution des systemes d'equations algebriques", school = "Universite P.etM. Curie", year = "1997", paper = "Maza97.pdf", keyword = "axiomref", comment = "\newline\refto{category NTSCAT NormalizedTriangularSetCategory}", url = "http://www.csd.uwo.ca/~moreno//Publications/MorenoMaza-Thesis-1997.ps.gz", abstract = "This thesis is dedicated to polynomial system solving by means of triangular sets. A first prt presents two algorithms to compute polynomial gcds over tower of simple extensions. The first one was designed by Renaud Rioboo and applies to algebraic towers. The second one is a generalization of the previous one to the most general case of seperable towers. These algorithms lead to an efficient implementation of two methods suggested by Daniel Lazard to solve polynomial systems by means of triangular sets. These programs solved problems that were previously unreachable. The second method was only sketched by its author. So, a second part of this thesis presents the necessary developements to describe a right implementation. Moreover, a theorecal and unified presentation, together with an experimental comparison with similar methods due to Wu Wen-Tsun, Dongming Wang and Michael Kalkbrener were realized by Philippe Aubry and are reported in a third part of this document." } \end{chunk} \index{Maza, Marc Moreno} \begin{chunk}{axiom.bib} @techreport{Maza00, author = "Maza, Marc Moreno", title = "On Triangular Decompositions of Algebraic Varieties", institution = "Numerical Algorithms Group", year = "2000", month = "June", type = "technical report", number = "TR 4/99", paper = "Maza00.pdf", url = "http://www.csd.uwo.ca/~moreno//Publications", keywords = "axiomref", abstract = "Different kinds of triangular decompositions of algebraic varieties are presented. The main result is an efficient method for obtaining them. Our strategy is based on a lifting theorem for polynomial computations module regular chains." } \end{chunk} committed Jun 24, 2016 7. Goal: Axiom Test Suite These are the equations from Dave89, Davenport's Looking at a set of equations'' paper. "This working paper describes our experiences with using the Groebner-basis method [Buchberger, 1985] to solve some related systems of polynomial equations. While we have not yet been able to solve the system that was our primary motivation, we feel that these experiences may prove useful to others investigating Buchberger's algorithm in this context, especially when, as is the case for the system under investigation, the equations are highly structured. We conclude with some examples of the polynomials that we factored in the course of this investigation." committed Jun 24, 2016 Commits on Jun 22, 2016 1. Goal: Axiom Literate Programming Collect algebra references in the bibliography committed Jun 22, 2016 2. Goal: Axiom Literate Programming Collect algebra references in the bibliography \index{Kalkbrener, M.} \begin{chunk}{axiom.bib} @phdthesis{Kalk91, author = "Kalkbrener, M.", title = "Three contributions to elimination theory", school = "University of Linz, Austria", year = "1991", comment = "\refto{category RSETCAT RegularTriangularSetCategory}" } \end{chunk} \begin{chunk}{axiom.bib} @misc{SALSA, title = "Solvers for Algebraic Systems and Applications", url = "http://www.ens-lyon.fr/LIP/Arenaire/SYMB/teams/salsa/proposal-salsa.pdf", comment = "\refto{category RSETCAT RegularTriangularSetCategory}", paper = "SALSA.pdf" } \end{chunk} \index{Hemmecke, Ralf} \begin{chunk}{axiom.bib} @phdthesis{Hemm03, author = "Hemmecke, Ralf", title = "Involutive Bases for Polynomial Ideals", school = "Johannes Kepler University, RISC", year = "2003", paper = "Hemm03.pdf", abstract = "This thesis contributes to the theory of polynomial involutive bases. Firstly, we present the two existing theories of involutive divisions, compare them, and come up with a generalised approach of {\sl suitable partial divisions}. The thesis is built on this generalized approach. Secondly, we treat the question of choosing a good'' suitable partial division in each iteration of the involutive basis algorithm. We devise an efficient and flexible algorithm for this purpose, the {\sl Sliced Division} algorithm. During the involutive basis algorithm, the Sliced Division algorithm contributes to an early detection of the involutive basis property and a minimisation of the number of critical elements. Thirdly, we give new criteria to avoid unnecessary reductions in an involutive basis algorithm. We show that the termination property of an involutive basis algorithm which applies our criteria is independent of the prolongation selection strategy used during its run. Finally, we present an implementation of the algorithm and results of this thesis in our software package CALIX." } \end{chunk} \index{Schorn, Markus} \begin{chunk}{axiom.bib} @phdthesis{Scho95, author = "Schorn, Markus", title = "Contributions to Symbolic Summation", school = "Johannes Kepler University, RISC", year = "1995", paper = "Scho95.pdf", url = "http://www.risc.jku.at/publications/download/risc_2246/diplom.pdf" } \end{chunk} \index{Winkler, Franz} \begin{chunk}{axiom.bib} @book{Wink96, author = "Winkler, Franz", title = "Polynomial Algorithms in Computer Algebra", year = "1996", publisher = "Springer-Verlag", isbn = "3.211-82759-5" } \end{chunk} \index{Buchberger, Bruno} \begin{chunk}{axiom.bib} @misc{Buch11, author = "Buchberger, Bruno", title = "Groebner Bases: A Short Introduction for System Theorists", year = "2011", abstract = "In this paper, we give a brief overview on Groebner bases theory, addressed to novices without prior knowledge in the field. After explaining the general strategy for solving problems via the Groebner approach, we develop the concept of Groebner bases by studying uniqueness of polynomial division (reduction''). For explicitly constructing Groebner bases, the crucial notion of S-polynomials is introduced, leading to the complete algorithmic solution of the construction problem. The algorithm is applied to examples from polynomial equation solving and algebraic relations. After a short discussion of complexity issues, we conclude the paper with some historical remarks and references." } \end{chunk} \index{Winkler, Franz} \begin{chunk}{axiom.bib} @article{Wink89, author = "Winkler, Franz", title = "Equational Theorem Proving and Rewrite Rule Systems", year = "1989", publisher = "Springer-Verlag", url = "http://www.risc.jku.at/publications/download/risc_3527/paper_47.pdf", paper = "Wink89.pdf", abstract = "Equational theorem proving is interesting both from a mathematical and a computational point of view. Many mathematical structures like monoids, groups, etc. can be described by equational axioms. So the theory of free monoids, free groups, etc. is the equational theory defined by these axioms. A decision procedure for the equational theory is a solution for the word problem over the associated algebraic structure. From a computational point of view, abstract data types are basically described by equations. Thus, proving properties of an abstract data type amounts to proving theorems in the associated equational theory. One approach to equational theorem proving consists in associating a direction with the equational axioms, thus transforming them into rewrite rules. Now in order to prove an equation$a=b$, the rewrite rules are applied to both sides, finally yielding reduced versions$a^{'}$and$b^{'}$of the left and right hand sides, respectively. If$a^{'}$and$b^{'}$agree syntactically, then the equation holds in the equational theory. However, in general this argument cannot be reversed;$a^{'}$and$b^{'}$might be different even if$a=b$is a theorem. The reason for this problem is that the rewrite system might not have the Church-Rosser property. So the goal is to take the original rewrite system and transform it into an equivalent one which has the desired Church-Rosser property. We show how rewrite systems can be used for proving theorems in equational and inductive theories, and how an equational specification of a problem can be turned into a rewrite program." } \end{chunk} \index{Collins, G.E.} \index{Mignotte, M.} \index{Winkler, F.} \begin{chunk}{axiom.bib} @article{Coll82, author = "Collins, G.E. and Mignotte, M. and Winkler, F.", title = "Arithmetic in Basic Algebraic Domains", publisher = "Springer-Verlag", journal = "Computing, Supplement 4", pages = "189-220", year = "1982", abstract = "This chapter is devoted to the arithmetic operations, essentially addition, multiplication, exponentiation, division, gcd calculations and evaluation, on the basic algebraic domains. The algorithms for these basic domains are those most frequently used in any computer algebra system. Therefore the best known algorithms, from a computational point of view, are presented. The basic domains considered here are the rational integers, the rational numbers, integers modulo$m$, Gaussian integers, polynomials, rational functions, power series, finite fields and$p$-adic numbers. BOunds on the maximum, minimum and average computing time ($t^{+},t^{-},t^{*}$) for the various algorithms are given." } \end{chunk} \index{Paule, Peter} \index{Kartashova, Lena} \index{Kauers, Manuel} \index{Schneider, Carsten} \index{Winkler, Franz} \begin{chunk}{axiom.bib} @misc{Paulxx, author = "Paule, Peter and Kartashova, Lena and Kauers, Manuel and Schneider, Carsten and Winkler, Franz", title = "Hot Topics in Symbolic Computation", publisher = "Springer", paper = "Paulxx.pdf", url = "http://www.risc.jku.at/publications/download/risc_3845/chapter1.pdf" } \end{chunk} \index{Johansson, Fredrik} \begin{chunk}{axiom.bib} @phdthesis{Joha14, author = "Johansson, Fredrik", title = "Fast and Rigorous Computation of Special Functions to High Precision", school = "Johannes Kepler University, Linz, Austria RISC", year = "2014", paper = "Joha14.pdf", abstract = "The problem of efficiently evaluating special functions to high precision has been considered by numerous authors. Important tools used for this purpose include algorithms for evaluation of linearly recurrent sequences, and algorithms for power series arithmetic. In this work, we give new baby-step, giant-step algorithms for evaluation of linearly recurrent sequences involving an expensive parameter (such as a high-precision real number) and for computing compositional inverses of power series. Our algorithms do not have the best asymptotic complexity, but they are faster than previous algorithms in practice over a large input range. Using a combination of techniques, we also obtain efficient new algorithms for numerically evaluating the gamma function$\Gamma(z)$and the Hurwitz zeta function$\zeta(s,a)$, or Taylor series expansions of those functions, with rigorous error bounds. Our methods achieve softly optimal complexity when computing a large number of derivatives to proportionally high precision. Finally, we show that isolated values of the integer partition function$p(n)$can be computed rigorously with softly optimal complexity by means of the Hardy-Ramanugan-Rademacher formula and careful numerical evaluation. We provide open source implementations which run significantly faster than previous published software. The implementations are used for record computations of the partition function, including the tabulation of several billion Ramanujan-type congruences, and of Taylor series associated with the Reimann zeta function." } \end{chunk} \index{Hodorog, Madalina} \begin{chunk}{axiom.bib} @phdthesis{Hodo11, author = "Hodorog, Madalina", title = "Symbolic-Numeric Algorithms for Plane Algebraic Curves", year = "2011", school = "RISC Research Institute for Symbolic Computation", paper = "Hodo11.pdf", abstract = "In computer algebra, the problem of computing topological invariants (i.e. delta-invariant, genus) of a plan complex algebraic curve is well-understood if the coefficients of the defining polynomial of the curve are exact data (i.e. integer numbers or rational numbers). The challenge is to handle this problem if the coefficients are inexact (i.e. numerical values). In this thesis, we approach the algebraic problem of computing invariants of a plane complex algebraic curve defined by a polynomial with both exact and inexact data. For the inexact data, we associate a positive real number called {\sl tolerance} or {\sl noise}, which measures the error level in the coefficients. We deal with an {\sl ill-posed} problem in the sense that, tiny changes in the input data lead to dramatic modifications in the output solution. For handling the ill-posedness of the problem we present a {\sl regularization} method, which estimates the invariants of a plane complex algebraic curve. Our regularization method consists of a set of {\sl symbolic-numeric algorithms} that extract structural information on the input curve, and of a {\sl parameter choice rule}, i.e. a function in the noise level. We first design the following symbolic-numeric algorithms for computing the invariants of a plane complex algebraic curve: \begin{itemize} \item we compute the link of each singularity of the curve by numerical equation solving \item we compute the Alexander polynomial of each link by using algorithms from computational geometry (i.e. an adapted version of the Bentley-Ottmann algorithm) and combinatorial objects from knot theory. \item we derive a formula for the delta-invariant and for the genus \end{itemize} We then prove that the symbolic-numeric algorithms together with the parameter choice rule compute approximate solutions, which satisfy the {\sl convergence for noisy data property}. Moreover, we perform several numerical experiments, which support the validity for the convergence statement. We implement the designed symbolic-numeric algorithms in a new software package called {\sl Genom3ck}, developed using the {\sl Axel} free algebraic modeler and the {\sl Mathemagix} free computer algebra system. For our purpose, both of these systems provide modern graphical capabilities, and algebraic and geometric tools for manipulating algebraic curves and surfaces defined by polynomials with both exact and inexact data. Together with its main functionality to compute the genus, the package {\sl Genom3ck} computes also other type of information on a plane complex algebraic curve, such as the singularities of the curve in the projective plane and the topological type of each singularity." } \end{chunk} \index{Er\"ocal, Bur\c{c}in} \begin{chunk}{axiom.bib} @phdthesis{Eroc11, author = {Er\"ocal, Bur\c{c}in}, title = "Algebraic Extensions for Symbolic Summation", school = "RISC Research Institute for Symbolic Computation", year = "2011", url = "http://www.risc.jku.at/publications/download/risc_4320/erocal_thesis.pdf", paper = "Eroc11.pdf", abstract = "The main result of this thesis is an effective method to extend Karr's symbolic summation framework to algebraic extensions. These arise, for example, when working with expressions involving$(-1)^n$. An implementation of this method, including a modernised version of Karr's algorithm is presented. Karr's algorithm is the summation analogue of the Risch algorithm for indefinite integration. In the summation case, towers of specialized difference fields called$\prod\sum$-fields are used to model nested sums and products. This is similar to the way elementary functions involving nested logarithms and exponentials are represented in differential fields in the integration case. In contrast to the integration framework, only transcendental extensions are allowed in Karr's construction. Algebraic extensions of$\prod\sum$-fields can even be rings with zero divisors. Karr's methods rely heavily on the ability to solve first-order linear difference equations and they are no longer applicable over these rings. Based on Bronstein's formulation of a method used by Singer for the solution of differential equations over algebraic extensions, we transform a first-order linear equation over an algebraic extension to a system of first-order equations over a purely transcendental extension field. However, this domain is not necessarily a$\prod\sum$-field. Using a structure theorem by Singer and van der Put, we reduce this system to a single first-order equation over a$\prod\sum$-field, which can be solved by Karr's algorithm. We also describe how to construct towers of difference ring extensions on an algebraic extension, where the same reduction methods can be used. A common bottleneck for symbolic summation algorithms is the computation of nullspaces of matrices over rational function fields. We present a fast algorithm for matrices over$\mathbb{Q}(x)\$
which uses fast arithmetic at the hardware level with calls to BLAS
subroutines after modular reduction. This part is joint work with Arne
Storjohann."
}

\end{chunk}
committed Jun 22, 2016
Commits on Jun 21, 2016
1. Goal: Axiom Literate Programming

Collect algebra references in the bibliography

\index{Schafer, R.D.}
\begin{chunk}{axiom.bib}
@article{Scha61,
author = "Schafer, R.D.",
title = "An Introduction to Nonassociative Algebras",
year = "1961",
comment = "\refto{category NARNG NonAssociativeRng}",
url = "http://www.gutenberg.org/ebooks/25156",
paper = "Scha61.pdf",
abstract =
"These are notes for my lectures in July, 1961, at the Advanced
Subject Matter Institute in Algebra which was held at Oklahoma State
University in the summer of 1961.

Students at the Institute were provided with reprints of my paper,
{\sl Structure and representation of nonassociate algebras} (Bulletin
of the American Mathematical Society, vol. 61 (1955), pp469-484),
together with copies of a selective bibliography of more recent papers
on non-associative algebras. These notes supplement the 1955 Bulletin
article, bringing the statements there up to date and providing
detailed proofs of a selected group of theorems. The proofs illustrate
a number of important techniques used in the study of nonassociative
algebras."
}

\end{chunk}

\index{Schafer, R.D.}
\begin{chunk}{axiom.bib}
@book{Scha66,
author = "Schafer, R.D.",
title = "An Introduction to Nonassociative Algebras",
year = "1966",
publisher = "Academic Press, New York",
comment = "\refto{category NARNG NonAssociativeRng}",
comment = "documentation for AlgebraGivenByStructuralConstants"

}

\end{chunk}

\index{Schafer, R.D.}
\begin{chunk}{axiom.bib}
@book{Scha10,
author = "Schafer, R.D.",
title = "An Introduction to Nonassociative Algebras",
year = "2010",
publisher = "Benediction Classics",
comment = "\refto{category NARNG NonAssociativeRng}",
isbn = "978-1849025904",
abstract =
"Concise study presents in a short space some of the important ideas
and results in the theory of non-associative algebras, with particular
emphasis on alternative and (commutative) Jordan algebras. Written as
an introduction for graduate students and other mathematicians meeting
the subject for the first time."
}

\end{chunk}

\index{Bremner, Murray R.}
\begin{chunk}{axiom.sty}
@misc{Brem08,
author = "Bremner, Murray R.",
title = "Nonassociative Algebras",
year = "2008",
comment = "\refto{category NARNG NonAssociativeRng}",
abstract =

"One of the earliest surveys on nonassociative algebras is the article
by Shirshov which introduced the phrase rings that are nearly
associative''. The first book in the English language devoted to a
systematic study of nonassociative algebras is Schafer (Scha66). A
comprehensive exposition of the work of the Russian School is
Zhevlakov, Slinko, Shestakov and Shirshov. A collection of open
research problems in algebra, including many problems on
nonassociative algebra, is the {\sl Dniester Notebook}; the survey
article by Kuzmin and Shetakov is from the same period. Three books on
Jordan algebras which contain substantial material on general
nonassociative algebras are Braun and Koecher, Jacobson, and
McCrimmon. Recent research appears in the Proceedings of the
International Conferences on Nonassociative Algebras and its
Applications. The present section provides very limited information on
Lie algebras, since they are the subject of Section 16.4. The last
part (section 9) of the present section presents three applications of
computational linear algebra to the study of polynomial identiies for
nonassociative algebras: pseudorandom vectors in a nonassociative
algebra, the expansion matrix for a nonassociative operation, and the
representation theory of the symmetric group."
}

\end{chunk}`
committed Jun 21, 2016
Something went wrong with that request. Please try again.