Computer Algebra Solving of First Order ODEs Using Symmetry Methods

A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the explicit determination of the coefficients of the infinitesimal symmetry generator; the construction of the most general invariant 1st. order ODE under given symmetries; the determination of the canonical coordinates of the underlying invariant group; and the testing of the returned results.Comment: 14 pages, LaTeX, submitted to Computer Physics Communications. Soft-package (On-Line Help) and sample MapleV session available at: http://dft.if.uerj.br/symbcomp.htm or ftp://dft.if.uerj.br/pdetool

Computer Algebra Solving of Second Order ODEs Using Symmetry Methods

An update of the ODEtools Maple package, for the analytical solving of 1st and 2nd order ODEs using Lie group symmetry methods, is presented. The set of routines includes an ODE-solver and user-level commands realizing most of the relevant steps of the symmetry scheme. The package also includes commands for testing the returned results, and for classifying 1st and 2nd order ODEs.Comment: 24 pages, LaTeX, Soft-package (On-Line help) and sample MapleV sessions available at: http://dft.if.uerj.br/odetools.htm or http://lie.uwaterloo.ca/odetools.ht

On the determination of cusp points of 3-R\underline{P}R parallel manipulators

This paper investigates the cuspidal configurations of 3-RPR parallel manipulators that may appear on their singular surfaces in the joint space. Cusp points play an important role in the kinematic behavior of parallel manipulators since they make possible a non-singular change of assembly mode. In previous works, the cusp points were calculated in sections of the joint space by solving a 24th-degree polynomial without any proof that this polynomial was the only one that gives all solutions. The purpose of this study is to propose a rigorous methodology to determine the cusp points of 3-R\underline{P}R manipulators and to certify that all cusp points are found. This methodology uses the notion of discriminant varieties and resorts to Gr\"obner bases for the solutions of systems of equations

Cusp Points in the Parameter Space of Degenerate 3-RPR Planar Parallel Manipulators

This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible non-singular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of Cylindric Algebraic Decomposition, Gr\"obner bases and Discriminant Varieties in order to partition the parameter space into cells with constant number of cusp points. These algorithms will allow us to classify a family of degenerate 3-RPR manipulators.Comment: ASME Journal of Mechanisms and Robotics (2012) 1-1

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial

Finding All Nash Equilibria of a Finite Game Using Polynomial Algebra

The set of Nash equilibria of a finite game is the set of nonnegative solutions to a system of polynomial equations. In this survey article we describe how to construct certain special games and explain how to find all the complex roots of the corresponding polynomial systems, including all the Nash equilibria. We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games. We describe the use of Groebner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions. Finally, we review the use of the Gambit software package to find all Nash equilibria of a finite game.Comment: Invited contribution to Journal of Economic Theory; includes color figure