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

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

Classification of 3-dimensional integrable scalar discrete equations

We classify all integrable 3-dimensional scalar discrete quasilinear equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An equation Q=0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces of the 4-dimensional lattice. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only nontrivial (non-linearizable) integrable equation from this class is the well-known dBKP-system. (Version 2: A small correction in Table 1 (p.7) for n=2 has been made.) (Version 3: A few small corrections: one more reference added, the main statement stated more explicitly.)Comment: 20 p. LaTeX + 1 EPS figur

Numerical Computing and Graphics for the Power Method Transformation Using Mathematica

This paper provides the requisite information and description of software that perform numerical computations and graphics for the power method polynomial transformation. The software developed is written in the Mathematica 5.2 package PowerMethod.m and is associated with fifth-order polynomials that are used for simulating univariate and multivariate non-normal distributions. The package is flexible enough to allow a user the choice to model theoretical pdfs, empirical data, or a user's own selected distribution(s). The primary functions perform the following (a) compute standardized cumulants and polynomial coefficients, (b) ensure that polynomial transformations yield valid pdfs, and (c) graph power method pdfs and cdfs. Other functions compute cumulative probabilities, modes, trimmed means, intermediate correlations, or perform the graphics associated with fitting power method pdfs to either empirical or theoretical distributions. Numerical examples and Monte Carlo results are provided to demonstrate and validate the use of the software package. The notebook Demo.nb is also provided as a guide for user of the power method.

A unified framework for solving a general class of conditional and robust set-membership estimation problems

In this paper we present a unified framework for solving a general class of problems arising in the context of set-membership estimation/identification theory. More precisely, the paper aims at providing an original approach for the computation of optimal conditional and robust projection estimates in a nonlinear estimation setting where the operator relating the data and the parameter to be estimated is assumed to be a generic multivariate polynomial function and the uncertainties affecting the data are assumed to belong to semialgebraic sets. By noticing that the computation of both the conditional and the robust projection optimal estimators requires the solution to min-max optimization problems that share the same structure, we propose a unified two-stage approach based on semidefinite-relaxation techniques for solving such estimation problems. The key idea of the proposed procedure is to recognize that the optimal functional of the inner optimization problems can be approximated to any desired precision by a multivariate polynomial function by suitably exploiting recently proposed results in the field of parametric optimization. Two simulation examples are reported to show the effectiveness of the proposed approach.Comment: Accpeted for publication in the IEEE Transactions on Automatic Control (2014