TY - GEN A1 - Wolf, Thomas T1 - The Parametric Solution of Underdetermined linear ODEs N2 - The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are arbitrary analytic functions in the independent variable. A second conceptual aim is to present an algorithm that is in some sense dual to the fundamental Euclids algorithm, and thus an alternative to the special case of a Gr\"{o}bner basis algorithm as it is used for solving linear ODE-systems. In the paper Euclids algorithm and the new dual version' are compared and their complementary strengths are analysed on the task of solving underdetermined ODEs. An implementation of the described algorithm is interactively accessible at http://lie.math.brocku.ca/crack/uode. T3 - ZIB-Report - 08-15 KW - linear ODEs KW - underdetermined systems KW - Euclid's algorithm KW - computer algebra KW - Reduce KW - Crack Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10693 SN - 1438-0064 ER - TY - GEN A1 - Tsarev, Sergey A1 - Wolf, Thomas T1 - Classification of 3-dimensional integrable scalar discrete equations N2 - We classify all integrable 3-dimensional scalar discrete affine linear equations $Q_3=0$ on an elementary cubic cell of the lattice ${\mathbb Z}^3$. An equation $Q_3=0$ %of such form is called integrable if it may be consistently imposed on all $3$-dimensional elementary faces of the lattice ${\mathbb Z}^4$. 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 ontrivial(non-linearizable) integrable equation from this class is the well-known dBKP-system. T3 - ZIB-Report - 08-13 KW - integrable systems KW - discrete equations KW - large polynomial systems KW - computer algebra KW - REDUCE KW - FORM KW - Crack Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10667 SN - 1438-0064 ER -