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On Groebner bases and their use in solving some practical problems
Matej Mencinger, 2013, original scientific article

Abstract: Groebner basis are an important theoretical building block of modern (polynomial) ring theory. The origin of Groebner basis theory goes back to solving some theoretical problems concerning the ideals in polynomial rings, as well as solving polynomial systems of equations. In this article four practical applications of Groebner basis theory are considered; we use Groebner basis to solve the systems of nonlinear polynomial equations, to solve an integer programming problem, to solve the problem of chromatic number of a graph, and finally we consider an original example from the theory of systems of ordinary (polynomial) differential equations. For practical computations we use systems MATHEMATICA and SINGULAR .
Keywords: polynomial system of (differential) equations, integer linear programming, chromatic number of a graph, polynomial rings, Groebner basis, CAS systems
Published in DKUM: 10.07.2015; Views: 1556; Downloads: 96
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The wavelet transform for BEM computational fluid dynamics
Jure Ravnik, Leopold Škerget, Matjaž Hriberšek, 2004, original scientific article

Abstract: A wavelet matrix compression technique was used to solve systems of linear equations resulting from BEM applied to fluid dynamics. The governing equations were written in velocity-vorticity formulation and solutions of the resulting systems of equations were obtained with and without wavelet matrix compression. A modification of the Haar wavelet transform, which can transformvectors of any size, is proposed. The threshold, used for making fully populated matrices sparse, was written as a product of a user defined factor and the average value of absolute matrix elements values. Numerical tests were performed to assert, that the error caused by wavelet compression depends linearly on the factor , while the dependence of the error on the share of thresholded elements in the system matrix is highly non-linear. The results also showed that the increasing non-linearity (higher Ra and Re numbervalues) limits the extent of compression. On the other hand, higher meshdensity enables higher compression ratios.
Keywords: fluid mechanics, computational fluid dynamics, boundary element method, wavelet transform, linear systems of equations, velocity vorticity formulation, driven cavity, natural convection, system matrix compression
Published in DKUM: 01.06.2012; Views: 2185; Downloads: 95
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