pinecone/core-2020-05-10-deduplication huggingface dataset:
Worksheets and Notebooks: Can We Teach Mathematical Algorithms with Them?

AbstractThis paper describes how computer algebra systems can be used to help students understand mathematical algorithms. Since one can write programs in these systems, one way to help the students is to show them complete programs and try to show them what the programs do. However, programs are like black boxes; you see what goes in and what comes out. To understand the program you need to show the steps that it makes because most students need to see a worked example in detail in order to see what an algorithm is doing before they understand how it works. Another problem with programs is that programs force you to be explicit about details; you must formulate the termination conditions, estimate degree bounds, specify domains of variables, etc. These details often get in the way of illustrating the basic algorithm.The premise of this article is that interactive worksheets and notebooks, which are now available in several computer algebra systems, sometimes provide a better alternative than programs as a teaching aid. In a worksheet or notebook, we can present an algorithm by following the sequence steps done. If part of the algorithm is difficult to express as a single command, or difficult to program, the results can be “typed in” instead. Other parts which may be difficult to formalize in a program, may be very obvious from looking at the output. To illustrate this, we have presented five algorithms as worksheets which we have used for teaching undergraduate mathematics and computer science classes
Computer Algebra and Interpolation: a Lesson Plan

AbstractPolynomial interpolation and related numerical analysis questions are used to take users of computer algebra systems from issuing one-liners to writing more complex programs. An alternate use of the outlined lessons is to introduce students to interpolation and acquaint them with splines and numerical differentiation. Constructive symbolic derivations of formulae for polynomial interpolants are presented. Numerical differentiation formulae are derived using them. Necessary background material is introduced. The examples and assignments can be done using any general purpose symbolic manipulator; complete codes using the computer algebra system AXIOM are given
Computing Isometries of Lattices

AbstractWe present the main ideas for an algorithm to calculate the group of automorphisms of a Euclidean lattice. This algorithm can be applied to related problems, e.g. to compute Bravais groups, to calculate automorphisms of lattices over number fields or, in a slightly modified version, to find isometries between lattices. An implementation of the algorithm by the second author has been successfully applied to lattices up to dimension 40 and allows, for example, obtaining of generators for the automorphism group of the Leech lattice in less than 30 min on a HP 9000/730 workstation
Analysing Finitely Presented Groups by Constructing Representations

AbstractWe present methods to construct representations of finitely presented groups. In well-conditioned examples it is possible to use Gröbner base and resultant methods to solve the system of algebraic equations obtained by evaluating the relations on matrices with indeterminates as entries. For more complicated cases we show how finite epimorphic images and their modular representation theory can be used to find good candidates for representations over a finite field that can be lifted to a local field.Finally, we apply lattice reduction methods to representations over local fields and obtain representations over algebraic number fields resp. rational function fields. Reducing modulo prime ideals thus gives infinitely many images of the finitely presented group
Computing with Subfields

AbstractThe algorithms presented here make use of subfield information to improve computations. For the three problems we deal with in this paper, the solution of relative norm equations, computation of independent units in the Galois closure of quartic fields and the computation of relative integral bases in quadratic extensions of number fields, we will outline the necessary algorithms that use subfield information and whenever possible we will compare the running time of each algorithm with the equivalent method not using subfield information
Integer Matrix Diagonalization

AbstractWe consider algorithms for computing the Smith normal form of integer matrices. A variety of different strategies have been proposed, primarily aimed at avoiding the major obstacle that occurs in such computations—explosive growth in size of intermediate entries. We present a new algorithm with excellent performance.We investigate the complexity of such computations, indicating relationships with NP-complete problems. We also describe new heuristics which perform well in practice. We present experimental evidence which shows our algorithm outperforming previous methods
Subexponential Algorithms for Class Group and Unit Computations

AbstractWe describe in detail the implementation of an algorithm which computes the class group and the unit group of a general number field, and solves the principal ideal problem. The basic ideas of this algorithm are due to J. Buchmann. New ideas are the use of LLL-reduction of an ideal in a given direction which replaces the notion of neighbour, and the use ofcomplexlogarithmic embeddings of elements which plays a crucial role. Heuristically the algorithm performs in sub-exponential time with respect to the discriminant for fixed degree, and performs well in practice
Increasing Model Building Capabilities by Constraint Solving on Terms with Integer Exponents

AbstractWe extend a former method for simultaneous search for refutations and models, based on the use of constraints, by extending the expressive power of the constraints. Our extension uses the language ofI-terms, in which it is possible to denote infinite sequences of structurally similar terms. Our work generalizes the results of Comon (1995). Comon gives only aunificationalgorithm for ``one hole''I-terms, while we present a decision procedure for term constraints with integer exponents (only decidability of the positive existential fragment was known so far). We show that the formalism ofI-terms can be profitably integrated into our method for model building. In particular, it allows us to specify all the resolvents of some kinds of self-resolvent clauses. This feature is especially useful in model building. We illustrate our approach with two non-trivial examples, showing that the extension presented here increases the power of our method, i.e. the class of formulae for which it is able to build models
Numeric Deduction in Symbolic Computation. Application to Normalizing
  Transformations

Algorithms of numeric (in exact arithmetic) deduction of analytical
expressions, proposed and described by Shevchenko and Vasiliev (1993), are
developed and implemented in a computer algebra code. This code is built as a
superstructure for the computer algebra package by Shevchenko and Sokolsky
(1993a) for normalization of Hamiltonian systems of ordinary differential
equations, in order that high complexity problems of normalization could be
solved. As an example, a resonant normal form of a Hamiltonian describing the
hyperboloidal precession of a dynamically symmetric satellite is derived by
means of the numeric deduction technique. The technique provides a considerable
economy, about 30 times in this particular application, in computer memory
consumption. It is naturally parallelizable. Thus the economy of memory
consumption is convertible into a gain in computation speed.Comment: 14 page
PSPCLink: A Cooperation Between General Symbolic and Poisson Series Processors

AbstractPoisson series appear frequently in problems of non-linear dynamics and celestial mechanics. The size of such mathematical objects makes their manipulation by means of general symbolic processors (GSP) inefficient. Special processors named Poisson series processors (PSP) have been created to handle them in a more efficient way. We propose here a way to combine the flexibility and easy use of GSPs with the power of PSPs. Using the communication toolMathLinkwe produced PSPCLink to connectMathematicawith PSPC, a PSP of our own creation. PSPCLink is a set of C-files and oneMathematicapackage. Compiling the C-files we created an executable program to be called fromMathematica. With PSPCLink we use the PSPC functions in a way completely integrated intoMathematicataking advantage of the good properties of bothMathematicaand PSPC
Computing the Newtonian Graph

AbstractIn his study of Newton's root approximation method, Smale (1985) defined the Newtonian graph of a complex univariate polynomialf. The vertices of this graph are the roots offandf′and the edges are the degenerate curves of flow of the Newtonian vector fieldNf(z) = −f(z)/f′(z). The embedded edges of this graph form the boundaries of root basins in Newton's root approximation method. The graph defines a treelike relation on the roots offandf′, similar to the linear order whenfhas only real roots.We give an efficient algebraic algorithm based on cell decomposition to compute the Newtonian graph. The resulting structure can be used to query whether two points in C are in the same basin. This suggests a modified version of Newton's method in which one can test whether a step has crossed a basin boundary. We show that this modified version does not necessarily converge to a root.Stefánsson (1995) has recently extended this algorithm to handle rational and algebraic functions without a significant increase in complexity. He has shown that the Newtonian graph tesselates the associated Riemann surface and can be used in conjunction with Euler's formula to give anNCalgorithm to calculate the genus of an algebraic curve
Nonlinear Control System Design by Quantifier Elimination

AbstractMany problems in control theory can be formulated as formulae in the first-order theory of real closed fields. In this paper we investigate some of the expressive power of this theory. We consider dynamical systems described by polynomial differential equations subjected to constraints on control and system variables and show how to formulate questions in the above framework which can be answered by quantifier elimination. The problems treated in this paper regard stationarity, stability, and following of a polynomially parametrized curve. The software package QEPCAD has been used to solve a number of examples
Robust Multi-Objective Feedback Design by Quantifier Elimination

AbstractThis paper shows how certain robust multi-objective feedback design problems can be reduced to quantifier elimination (QE) problems. In particular it is shown how robust stabilization and robust frequency domain performance specifications can be reduced to systems of polynomial inequalities with suitable logic quantifiers, ∀ and ∃. Because of computational complexity the size of problems that can solved by QE methods is limited. However, the design problems considered here do not haveanalyticalsolutions, so that even the solution of modest-sized problems may be of practical interest
Testing Stability by Quantifier Elimination

AbstractFor initial and initial-boundary value problems described by differential equations, stability requires the solutions to behave well for large times. For linear constant-coefficient problems, Fourier and Laplace transforms are used to convert stability problems to questions about roots of polynomials. Many of these questions can be viewed, in a natural way, as quantifier-elimination problems. The Tarski–Seidenberg theorem shows that quantifier-elimination problems are solvable in a finite number of steps. However, the complexity of this algorithm makes it impractical for even the simplest problems. The newer Quantifier Elimination by Partial Algebraic Decomposition (QEPCAD) algorithm is far more practical, allowing the solution of some non-trivial problems. In this paper, we show how to write all common stability problems as quantifier-elimination problems, and develop a set of computer-algebra tools that allows us to find analytic solutions to simple stability problems in a few seconds, and to solve some interesting problems in from a few minutes to a few hours
Simplification of Quantifier-free Formulae over Ordered Fields

AbstractGiven a quantifier-free first-order formula over the theory of ordered fields, our aim is to find an equivalent first-order formula that issimpler. The notion of a formula being simpler will be specified. An overview is given over various methods combining elements of field theory, order theory, and logic. These methods do not require a Boolean normal form computation. They have been developed and implemented inreducefor simplifying intermediate and final results of automatic quantifier elimination by elimination sets. Their applicability is certainly not restricted to the area of quantifier elimination
Lattices of Compatibly Embedded Finite Fields

AbstractThe design of a computational facility for finite fields that allows complete freedom in the manner in which fields are constructed, is complicated by the fact that a field of fixed isomorphism typeKmay be constructed in many different ways. It is desirable that the user be able to perform simultaneous computations in different versions ofKin such a way that isomorphisms identifying elements in the different versions are applied automatically whenever necessary. This paper presents a coherent scheme for solving this problem based on an efficient method for compatibly embedding one field within another. This scheme forms a central component of theMAGMAmodule for finite fields. The paper also outlines the different representations of finite fields employed in the package and comments briefly on some of the major algorithms
Constructing Permutation Representations for Matrix Groups

AbstractNew techniques, both theoretical and practical, are presented for constructing permutation representations for computing with matrix groups defined over finite fields. The permutation representation is constructed on a conjugacy class of subgroups of prime order. We construct a base for the permutation representation, which in turn simplifies the computation of a strong generating set. In addition, we present an elementary test for checking the simplicity of the permutation image. The theory has been successfully tested on a representation of the sporadic simple groupLy, discovered by Lyons(1972). With noa prioriassumptions, we find a permutation representation of degree 9606125 on a conjugacy class of subgroups of order 3, find the order of the resulting permutation group, and verify simplicity. A Monte Carlo variation of the algorithm was used to achieve better space and time efficiency. The construction of the permutation representation required four CPU days on a SPARCserver 670MP with 64 MB. The permutation representation was used implicitly in the sense that the group element was stored as a matrix, and its permutation action on a “point” was determined using a pre-computed data structure. Thus, additional computations required little additional space. The algorithm has also been implemented using the MasPar MP-1 SIMD parallel computer and 8 SPARC-2's running under MPI. The results of those parallel experiments are briefly reviewed
On Testing a Bivariate Polynomial for Analytic Reducibility

AbstractLetKbe an algebraically closed field of characteristic zero. We present an efficient algorithm for determining whether or not a given polynomialf(x,y) inK[x,y] is analytically reducible overKat the origin. The algorithm presented is based upon an informal method sketched by Kuo (1989) which is in turn derived from ideas of Abhyankar (1988). The presentation contained herein emphasises the proofs of the algorithm's correctness and termination, and is suitable for computer implementation. A polynomial worst case time complexity bound is proved for a partial version of the algorithm
Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations

AbstractA new algorithm for the symbolic computation of polynomial conserved densities for systems of nonlinear evolution equations is presented. The algorithm is implemented inMathematica. The programcondens.mautomatically carries out the lengthy symbolic computations for the construction of conserved densities. The code is tested on several well-known partial differential equations from soliton theory. For systems with parameters,condens.mcan be used to determine the conditions on these parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor for integrability of the system
How to Recognize Zero

AbstractAn elementary point is a point in complexnspace, which is an isolated, nonsingular solution ofnequations innvariables, each equation being either of the formp=0, wherepis a polynomial in Q[x1,…,xn], or of the formxj−exi=0. An elementary number is the polynomial image of an elementary point. In this article a semi algorithm is given to decide whether or not a given elementary number is zero. It is proved that this semi algorithm is an algorithm, i.e. that it always terminates, unless it is given a problem containing a counterexample to Schanuel’s conjecture
On Validated Computing in Algebraic Number Fields

AbstractIn this paper we discuss several computations with elements of algebraic number fields F which either require numerical calculations or are at least speeded up considerably by their use. Since the results can be uniquely represented by (sequences of) rational integers, those numerical calculations should clearly be performed by validated methods. Hence, the intention of this paper is to point out important number theoretical problems which can be adequately solved only by the use of symbolic and validated computations
Floating-point versus Symbolic Computations in theQD-algorithm

AbstractThe convergence of columns in the univariateqd-algorithm to reciprocals of polar singularities of meromorphic functions has often proved to be very useful. Anyq-column corresponding to a “simple pole of isolated modulus” converges to the reciprocal of the corresponding pole. By performing an equivalence transformation of the underlying corresponding continued fraction and programming the newqd-like scheme so as to compute algebraic expressions, the difference in convergence behaviour between the “simple pole” case and the “equal modulus” pole case of the floating-point algorithm is eliminated
A New Approach for Analytical Computation of Hamiltonian of a Satellite Perturbed Motion

AbstractThe problem of computation of the Hamiltonian for the perturbed motion of an artificial Earth satellite with large eccentricity is considered. The method of calculation of items of the perturbing function, which is presented in the form of Jacobi nome series, is described. Some advantages of the new approach to the expansion of the Hamiltonian are shown by computations within computer algebra systems. Mathematical techniques for computations with elliptic functions, which are used in the suggested method of Hamiltonian expansion, are considered. Numerical verification and accuracy checking of analytical results is investigated by means of computer algebra system MAPLE
Counting Points on Curves over Finite Fields

AbstractWe consider the problem of counting the number of points on a plane curve, defined by a homogeneous polynomialF(x,y,z) ∈Fq[x,y,z], which are rational over a ground field Fq. More precisely, we show that if we are given a projective plane curve C of degreen, and if C has only ordinary multiple points, then one can compute the number of Fq-rational points on C in randomized time (logq)Δwhere Δ=nO(1). Since our algorithm actually computes the characteristic polynomial of the Frobenius endomorphism on the Jacobian of C, it ,follows that we may also compute (1) the number of Fq-rational points on the smooth projective model of C, (2) the number of Fq-rational points on the Jacobian of C, and (3) the number of Fqm-rational points on C in any given finite extension Fqmof the ground field, each in a similar time bound
A Generalization of Gröbner Basis Algorithms to Polycyclic Group Rings

AbstractIt is well-known that for the integral group ring of a polycyclic group several decision problems are decidable, in particular the ideal membership problem. In this paper we define an effective reduction relation for group rings over polycyclic groups. This reduction is based on left multiplication and hence corresponds to left ideals. Using this reduction we present a generalization of Buchberger's Gröbner basis method by giving an appropriate definition of “Gröbner bases” in this setting and by characterizing them using the concepts of saturation and s-polynomials. The approach is extended to two-sided ideals and a discussion on a Gröbner bases approach for right ideals is included
A Survey of User Interfaces for Computer Algebra Systems

AbstractThis paper surveys work within the Computer Algebra community (and elsewhere) directed towards improving user interfaces for scientific computation during the period 1963–1994. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. Work related to improving human interaction with computer algebra systems is the main focus of the paper. However, the paper includes additional materials on some closely related issues such as structured document editing, graphics, and communication protocols
A Generic Approach to Building User Interfaces for Theorem Provers

AbstractIn this paper, we present the results of an ongoing effort in building user interfaces for proof systems. Our approach is generic: we are not constructing a user interface for a particular proof system, rather we have developed techniques and tools that have been applied to several proof systems. We first propose and motivate a distributed architecture, where the proof system and the interface are two separate processes communicating through a protocol. Then we describe three high-level features:proof-by-pointing,script management, andtextual explanation. Altogether, they take advantage of the underlying architecture and yield a more user-friendly proof environment
NEWTON: An Interactive Environment for Exploring Mathematics

AbstractWe have developed a unique computational environment for use in teaching introductory mathematics. Our system, calledNewton, runs on Macintosh computers and consists of a user-friendly interface to the symbolic mathematics package Maple, supplemented by an extensive library of our own Maple code. Formulas are easily constructed and modified, appearing like those in textbooks; multiple windows allow users to see and work with several formulas at once. Formulas, graphs, and text can be intermixed in collapsible sections on worksheets. Users do not interact with Maple directly and need know nothing of Maple's syntax and command structure. Mathematical operations are selected from menus, with the added bonus that this makes it possible to document solutions. Originally designed for use in conjunction with the two semester introductory calculus sequence, the system has been extended for multivariate and vector calculus, linear algebra, numerical analysis, and differential equations