DSpace Collection: Volume 1 Number 4
http://hdl.handle.net/10525/319
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A Computer Algebra Application to Determination of Lie Symmetries of Partial Differential Equations
http://hdl.handle.net/10525/359
Title: A Computer Algebra Application to Determination of Lie Symmetries of Partial Differential Equations<br/><br/>Authors: Pulov, Vladimir; Chacarov, Edy; Uzunov, Ivan<br/><br/>Abstract: A MATHEMATICA package for finding Lie symmetries ofpartial differential equations is presented. The package is designed to createand solve the associated determining system of equations, the full set ofsolutions of which generates the widest permissible local Lie group of pointsymmetry transformations. Examples illustrating the functionality of thepackage's tools are given. The results of the package application to performinga full Lie group analysis of coupled nonlinear Schrödinger equations fromnonlinear fiber optics are presented. Comparisons with earlier publishedcomputer algebra implementations of the Lie group method are discussed.<br/><br/>Description: The paper has been presented at the 12th International Conference on Applications ofComputer Algebra, Varna, Bulgaria, June, 2006Introduction to the Maple Power Tool Intpakx
http://hdl.handle.net/10525/358
Title: Introduction to the Maple Power Tool Intpakx<br/><br/>Authors: Krämer, Walter<br/><br/>Abstract: The Maple Power Tool intpakX [24] de nes Maple types forreal intervals and complex disc intervals. On the level of basic operations,intpakX includes the four basic arithmetic operators, including extendedinterval division as an extra function. Furthermore, there are power, square,square root, logarithm and exponential functions, a set of standard functions,union, and intersection. Reimplementations of the Maple construction,conversion, and unapplication functions are available. Additionally, thereis a range of operators for complex disc arithmetic.As applications, verified computation of zeroes (Interval Newton Me-thod) with the possibility to find all zeroes of a function on a specifiedinterval, and range enclosure for real-valued functions of one or two variablesare implemented, the latter using either interval evaluation or evaluationvia the mean value form and adaptive subdivision of intervals. The usercan choose between a non-graphical and a graphical version of the abovealgorithms displaying the resulting intervals of each iteration step.The source code (about 2000 lines of Maple{code) of the extensionintpakX is freely available [23].<br/><br/>Description: The paper has been presented at the 12th International Conference on Applications ofComputer Algebra, Varna, Bulgaria, June, 2006Computing and Visualizing Solution Sets of Interval Linear Systems
http://hdl.handle.net/10525/357
Title: Computing and Visualizing Solution Sets of Interval Linear Systems<br/><br/>Authors: Krämer, Walter<br/><br/>Abstract: The computation of the exact solution set of an interval linearsystem is a nontrivial task [2, 13]. Even in two and three dimensions a lot ofwork has to be done. We demonstrate two different realizations. The firstapproach (see [16]) is based on Java, Java3D, and the BigRational package[21]. An applet allows modifications of the matrix coefficients and/or thecoefficients of the right hand side with concurrent real time visualization ofthe corresponding solution sets. The second approach (see [5]) uses Mapleand intpakX [22, 8, 12] to implement routines for the computation andvisualization of two and three dimensional solution sets. The regularity ofthe interval matrix A is verified by showing that ρ(|I-mid^(-1)(A)*Aj|) < 1[14]. Here, I means the identity matrix, mid(A) denotes the midpoint matrixand ρ denotes the spectral radius of a real matrix.<br/><br/>Description: The paper has been presented at the 12th International Conference on Applications ofComputer Algebra, Varna, Bulgaria, June, 2006Algebraic Computations with Hausdorff Continuous Functions
http://hdl.handle.net/10525/356
Title: Algebraic Computations with Hausdorff Continuous Functions<br/><br/>Authors: Anguelov, Roumen<br/><br/>Abstract: The set of Hausdorff continuous functions is the largest set ofinterval valued functions to which the ring structure of the set of continuousreal functions can be extended. The paper deals with the automation ofthe algebraic operations for Hausdorff continuous functions using an ultra-arithmetical approach.<br/><br/>Description: The paper has been presented at the 12th International Conference on Applications ofComputer Algebra, Varna, Bulgaria, June, 2006.Optimization of Rational Approximations by Continued Fractions
http://hdl.handle.net/10525/355
Title: Optimization of Rational Approximations by Continued Fractions<br/><br/>Authors: Blomquist, Frithjof<br/><br/>Abstract: To get guaranteed machine enclosures of a special function f(x),an upper bound ε(f) of the relative error is needed, where ε(f) itself dependson the error bounds ε(app); ε(eval) of the approximation and evaluation errorrespectively. The approximation function g(x) ≈ f(x) is a rational function(Remez algorithm), and with sufficiently high polynomial degrees ε(app)becomes sufficiently small. Evaluating g(x) on the machine produces arather great ε(eval) because of the division of the two erroneous polynomials.However, ε(eval) can distinctly be decreased, if the rational function g(x)is substituted by an appropriate continued fraction c(x) which in generalneeds less elementary operations than the original rational function g(x).Numerical examples will illustrate this advantage.<br/><br/>Description: The paper has been presented at the 12th International Conference on Applications ofComputer Algebra, Varna, Bulgaria, June, 2006.A Relation between the Weyl Group W(e8) and Eight-Line Arrangements on a Real Projective Plane
http://hdl.handle.net/10525/354
Title: A Relation between the Weyl Group W(e8) and Eight-Line Arrangements on a Real Projective Plane<br/><br/>Authors: Fukui, Tetsuo; Sekiguchi, Jiro<br/><br/>Abstract: The Weyl group W(E8) acts on the con guration space ofsystems of labelled eight lines on a real projective plane. With a system ofeight lines with a certain condition, a diagram consisting of ten roots of theroot system of type E8 is associated. We have already shown the existenceof a W(E8)-equivariant map of the totality of such diagrams to the set ofsystems of labelled eight lines. The purpose of this paper is to report thatthe map is injective.<br/><br/>Description: The paper has been presented at the 12th International Conference on Applications ofComputer Algebra, Varna, Bulgaria, June, 2006.Symbolic Dynamics in the Free-Fall Equal-Mass Three-Body Problem
http://hdl.handle.net/10525/353
Title: Symbolic Dynamics in the Free-Fall Equal-Mass Three-Body Problem<br/><br/>Authors: Mylläri, Aleksandr; Martynova, Alija; Orlov, Victor; Chernin, Arthur<br/><br/>Abstract: Free-fall equal-mass three-body systems are numerically studiedusing symbolic dynamics. We scan the two-dimensional homology map ofinitial configurations in steps of 0.001 along both axes. States of binary andtriple encounters as well as changes of configuration are used to constructsymbolic sequences. Symbolic sequences are characterized by Shannon andMarkov entropies. Different ergodic components corresponding to differentdistinct peaks on the histograms of these entropies are revealed.<br/><br/>Description: The paper has been presented at the 12th International Conference on Applications ofComputer Algebra, Varna, Bulgaria, June, 2006Solving Differential Equations by Parallel Laplace Method with Assured Accuracy
http://hdl.handle.net/10525/352
Title: Solving Differential Equations by Parallel Laplace Method with Assured Accuracy<br/><br/>Authors: Malaschonok, Natasha<br/><br/>Abstract: We produce a parallel algorithm realizing the Laplace transformmethod for the symbolic solving of differential equations.In this paper we consider systems of ordinary linear differential equationswith constant coefficients, nonzero initial conditions and right-hand partsreduced to sums of exponents with polynomial coefficients.<br/><br/>Description: The paper has been presented at the 12th International Conference on Applications ofComputer Algebra, Varna, Bulgaria, June, 2006