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Systems of word equations, (at least "nontrivial" ones), express the fact that a certain finite set of words, (namely the images () of the unknowns ), satisfy some nontrivial relation(s). So it can be said that systems S {\displaystyle S} of word equations cause a defect effect in the sets of words X = { h ( x ) : x ∈ Ξ } {\displaystyle X ...
The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
The consequence of this difference is that at every step, a system of algebraic equations has to be solved. This increases the computational cost considerably. If a method with s stages is used to solve a differential equation with m components, then the system of algebraic equations has ms components.
"New high-order Runge-Kutta formulas with step size control for systems of first and second-order differential equations". Zeitschrift für Angewandte Mathematik und Mechanik . 44 (S1): T17 – T29 .
On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems (PDF) (Thesis). Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0.
The following tables provide a comparison of computer algebra systems (CAS). [ 1 ] [ 2 ] [ 3 ] A CAS is a package comprising a set of algorithms for performing symbolic manipulations on algebraic objects, a language to implement them, and an environment in which to use the language.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
SAS, [30] a system of software products for statistics. It includes SAS/IML, [31] a matrix programming language. VisSim is a visual block-diagram language for simulation of nonlinear dynamic systems and model-based embedded development. Its fast ODE engine supports real-time simulation of complex large-scale models.