Skip to content
Commits on Jun 29, 2016
  1. books/bookvolbib Axiom Citations in the Literature

    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. src/input/Makefile fix typo

    Goal: Axiom build
    
    Somewhere along the way I fat-fingered a character delete
    causing the build to break. Sigh.
    committed Jun 28, 2016
  2. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvolbib Axiom Citations in the Literature

    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. buglist: add todo 339: missing side conditions

    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. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvol* add Mulders to ORE and LODO refs

    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. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvol* add Abramov and Bronstein reference to LODO and ORE

    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. books/bookvol* add Bronstein LODO and ORE reference

    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. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvol10.3 Add Riob92, Emir04 domain RECLOS RealClosure

    Goal: Axiom Literate Programming
    committed Jun 26, 2016
  9. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvol10.4 add Rube06, Hebi10 GUESS references

    Goal: Axiom Literate Programming
    committed Jun 25, 2016
  11. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvol* standardize the bibliography/index sections

    Goal: Axiom Literate Programming
    committed Jun 25, 2016
  2. books/bookvol9 add Spad BNF syntax from Smit07

    Goal: Axiom Literate Programming
    committed Jun 25, 2016
  3. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvolbib Axiom Citations in the Literature

    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. src/input/quantunwalk.input Code for quantum walk from McGettrick

    Goal: Axiom Test Code
    committed Jun 25, 2016
  6. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvolbib Axiom Citations in the Literature

    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. books/bookvolbib Arna95, Dave92, Jaes93 package IntegerPrimesPackage

    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. books/bookvolbib Add Axiom Citations in the Literature

    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. books/bookvolbib Lamb92 category FMCAT FreeModuleCat

    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. books/bookvolbib Aubr96 package DFSFUN DoubleFloatSpecialFunctions refs

    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. books/bookvolbib category NTSCAT NormalizedTriangularSetCategory refs

    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. src/input/dave89 Davenport Looking at a set of equations

    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. books/bookvolbib category RSETCAT RegularTriangularSetCategory refs

    Goal: Axiom Literate Programming
    
    Collect algebra references in the bibliography
    committed Jun 22, 2016
  2. books/bookvolbib add RISC references

    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. books/bookvolbib Scha61,Scha66,Scha10,Brem08 category NonAssociativeRng

    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.